geometric sequence problem solving with solutions

Geometric Sequences Problems with Solutions

Geometric sequences are used in several branches of applied mathematics to engineering, sciences, computer sciences, biology, finance... Problems and exercises involving geometric sequences, along with answers are presented.

Review OF Geometric Sequences

The sequence shown below

Problems with Solutions

Problem 1 Find the terms a 2 , a 3 , a 4 and a 5 of a geometric sequence if a 1 = 10 and the common ratio r = - 1. Solution to Problem 1: Use the definition of a geometric sequence $ a_2 = a_1 \times r = 10 (-1) = - 10 \\ a_3 = a_2 \times r = - 10 (-1) = 10 \\ a_4 = a_3 \times r = 10 (-1) = - 10 \\ a_5 = a_4 \times r = - 10 (-1) = 10 $

Find the 10 th term of a geometric sequence if a 1 = 45 and the common ration r = 0.2. Solution to Problem 2: Use the formula \[ a_n = a_1 \times r^{n-1} \] that gives the n th term to find a 10 as follows $ a_{10} = 45 \times 0.2^{10-1} = 2.304 \times 10^{-5} $

Find a 20 of a geometric sequence if the first few terms of the sequence are given by

Given the terms a 10 = 3 / 512 and a 15 = 3 / 16384 of a geometric sequence, find the exact value of the term a 30 of the sequence. Solution to Problem 4: We first use the formula for the n th term to write a 10 and a 15 as follows $ a_{10} = a_1 \times r^{10-1} = a_1 r^9 = 3 / 512 \\ \\ a_{15} = a_1 \times r^{15-1} = a_1 r^{14} = 3 / 16384 $ We now divide the terms a 10 and a 15 to write $ a_{15} / a_{10} = a_1 \times r^{14} / (a_1 \times r^9) = (3 / 16384) / (3 / 512) $ Simplify expressions in the above equation to obtain. r 5 = 1 / 32 which gives r = 1/2 We now use a 10 to find a 1 as follows. $ a_{10} = 3 / 512 = a_1 (1/2)^9 $ Solve for a 1 to obtain. $ a_1 = 3 $ We now use the formula for the n th term to find a 30 as follows. $ a_{30} = 3(1/2)^{29} = 3 / 536870912 $

Find the sum \[ S = \sum_{k=1}^{6} 3^{k - 1} \] Solution to Problem 5: We first rewrite the sum S as follows S = 1 + 3 + 9 + 27 + 81 + 243 = 364 Another method is to first note that the terms making the sum are those of a geometric sequence with a 1 = 1 and r = 3 using the formula s n = a 1 (1 - r n ) / (1 - r) with n = 6. s 6 = 1 (1 - 3 6 ) / (1 - 3) = 364

Find the sum \[ S = \sum_{i=1}^{10} 8 \times (1/4)^{i - 1} \] Solution to Problem 6: An examination of the terms included in the sum are 8 , 8× ((1/4) 1 , 8×((1/4) 2 , ... , 8×((1/4) 9 These are the terms of a geometric sequence with a 1 = 8 and r = 1/4 and therefore we can use the formula for the sum of the terms of a geometric sequence s 10 = a 1 (1 - r n ) / (1 - r) = 8 × (1 - (1/4) 10 ) / (1 - 1/4) = 10.67 (rounded to 2 decimal places)

Write the rational number 5.31313131... as the ratio of two integers. Solution to Problem 7: We first write the given rational number as an infinite sum as follows 5.313131... = 5 + 0.31 + 0.0031 + 0.000031 + .... The terms making 0.31 + 0.0031 + 0.000031 ... are those of a geometric sequence with a 1 = 0.31 and r = 0.01. Hence the use of the formula for an infinite sum of a geometric sequence S = a 1 / (1 - r) = 0.31 / (1 - 0.01) = 0.31 / 0.99 = 31 / 99 We now write 5.313131... as follows 5.313131... = 5 + 31/99 = 526 / 99

Exercises with Answers

Answer the following questions related to geometric sequences: a) Find a 20 given that a 3 = 1/2 and a 5 = 8 b) Find a 30 given that the first few terms of a geometric sequence are given by -2 , 1 , -1/2 , 1/4 ... c) Find r given that a 1 = 10 and a 20 = 10 -18 d) write the rational number 0.9717171... as a ratio of two positive integers.

a) a 20 = 2 18 b) a 30 = 1 / 2 28 c) r = 0.1 d) 0.9717171... = 481/495

How to Solve Geometric Sequences? (+FREE Worksheet!)

Learn how to solve Geometric Sequence problems using the following step-by-step guide with detailed solutions.

Step by step guide to solve Geometric Sequence Problems

It is a sequence of numbers where each term after the first is found by multiplying the previous item by the common ratio, a fixed, non-zero number. For example, the sequence $2, 4, 8, 16, 32$, … is a geometric sequence with a common ratio of $2$.
To find any term in a geometric sequence use this formula: $\color{blue}{x_{n}=ar^{(n – 1)}}$
$a =$ the first term , $r =$ the common ratio , $n =$ number of items

Geometric Sequences – Example 1:

Given the first term and the common ratio of a geometric sequence find the first five terms of the sequence. $a_1=3,r=-2$

Use geometric sequence formula: $\color{blue}{x_{n}=ar^{(n – 1)}}$ $→x_{n}=0.8 .(-5)^{n-1}$ If $n=1$ then: $x_{1}=3 .(-2)^{1-1}=3 (1)=3$, First Five Terms: $3,-6,12,-24,48$

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Geometric sequences – example 2:.

Given two terms in a geometric sequence find the 8th term. $a_{3}=10$ and $a_{5}=40$

Use geometric sequence formula: $\color{blue}{x_{n}=ar^{(n – 1)}}$ $→a_{3}=ar^{(3 – 1)}=ar^2=10$ $x_{n}=ar^{(n – 1)}→a_5=ar^{(5 – 1)}=ar^4=40$ Now divide $a_{5}$ by $a_{3}$. Then: $\frac{a_{5}}{a_{3}} =\frac{ar^4}{ar^2 }=\frac{40}{10}$, Now simplify: $\frac{ar^4}{ar^2 }=\frac{40}{10}→r^2=4→r=2$ We can find a now: $ar^2=12→a(2^2 )=10→a=2.5$ Use the formula to find the 8th term: $x_{n}=ar^{(n – 1)}→a_8=(2.5) (2)^8=2.5(256)=640$

Geometric Sequences – Example 3:

Given the first term and the common ratio of a geometric sequence find the first five terms of the sequence. $a_{1}=0.8,r=-5$

Use geometric sequence formula: $\color{blue}{x_{n}=ar^{(n – 1)}}$ $→x_{n}=0.8 .(-5)^{n-1}$ If $n=1$ then: $x_{1}=0.8 .(-5)^{1-1}=0.8 (1)=0.8$, First Five Terms: $0.8,-4,20,-100,500$

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Geometric sequences – example 4:.

Given two terms in a geometric sequence find the 8th term. $a_3=12$ and $a_5=48$

Use geometric sequence formula: $\color{blue}{x_{n}=ar^{(n – 1)}}$ $→a_3=ar^{(3 – 1)}=ar^2=12$ $\color{blue}{x_{n}=ar^{(n – 1)}}$ $→a_5=ar^{(5 – 1)}=ar^4=48$ Now divide $a_{5}$ by $a_{3}$. Then: $\frac{a_{5}}{a_{3} }=\frac{ar^4}{ar^2}=\frac{48}{12}$, Now simplify: $\frac{ar^4}{ar^2}=\frac{48}{12}→r^2=4→r=2$ We can find a now: $ar^2=12→a(2^2 )=12→a=3$ Use the formula to find the $8^{th}$ term: $\color{blue}{x_{n}=ar^{(n – 1)}}$ $→a_{8}=(3) (2)^8=3(256)=768$

Exercises for Solving Geometric Sequences

Determine if the sequence is geometric. if it is, find the common ratio..

$\color{blue}{1, – 5, 25, – 125, …}$
$\color{blue}{– 2, – 4, – 8, – 16, …}$
$\color{blue}{4, 16, 36, 64, …}$
$\color{blue}{– 3, – 15, – 75, – 375, …}$

Download Geometric Sequences Worksheet

$\color{blue}{r=-5}$
$\color{blue}{r=2}$
$\color{blue}{not \ geometric}$
$\color{blue}{r=5}$

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Geometric Sequences

Here we will learn what a geometric sequence is, how to continue a geometric sequence, how to find missing terms in a geometric sequence, and how to generate a geometric sequence.

At the end, you’ll find geometric sequence worksheets based on Edexcel, AQA, and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is a geometric sequence?

A geometric sequence (geometric progression) is an ordered set of numbers that progresses by multiplying or dividing each term by a common ratio.

If we multiply or divide by the same number each time to make the sequence, it is a geometric sequence .

The common ratio is the same for any two consecutive terms in the same sequence.

Here are a few examples,

What are geometric sequences?

Geometric sequence formula

The geometric sequence formula is,

Where,

\pmb{ a_{n} } is the n^{th} term (general term),

\pmb{ a_{1} } is the first term,

\pmb{ n } is the term position,

and \pmb{ r } is the common ratio.

We get the geometric sequence formula by looking at the following example,

We can see the common ratio (r) is 2 , so r = 2 .

a_{1} is the first term which is 5 ,

a_{2} is the second term which is 10 ,

and a_{3} is the third term which is 20 etc.

However we can write this using the common difference of 2 ,

geometric sequence problem solving with solutions

Related lessons on sequences

Geometric sequences is part of our series of lessons to support revision on sequences . You may find it helpful to start with the main sequences lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Quadratic sequences
Arithmetic sequence
Nth term of a sequence
Recurrence relation
Quadratic nth term

How to continue a geometric sequence

To continue a geometric sequence, you need to calculate the common ratio. This is the factor that is used to multiply one term to get the next term. To calculate the common ratio and continue a geometric sequence you need to:

Take two consecutive terms from the sequence.

Divide the second term by the first term to find the common ratio r .

Multiply the last term in the sequence by the common ratio to find the next term. Repeat for each new term.

Explain how to continue a geometric sequence

Geometric sequences worksheet

Get your free geometric sequences worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Continuing a geometric sequence examples

Example 1: continuing a geometric sequence.

Calculate the next three terms for the geometric progression 1, 2, 4, 8, 16, …

Here we will take the numbers 4 and 8 .

2 Divide the second term by the first term to find the value of the common ratio, r .

3 Multiply the last term in the sequence by the common ratio to find the next term. Repeat for each new term.

The next three terms in the sequence are 32, 64, and 128 .

Example 2: continuing a geometric sequence with negative numbers

Calculate the next three terms for the sequence -2, -10, -50, -250, -1250, …

Here we will take the numbers -10 and -50 .

Divide the second term by the first term to find the common ratio, r.

The next three terms are -6250, -31250, and -156250.

Example 3: continuing a geometric sequence with decimals

Calculate the next three terms for the sequence 100, 10, 1, 0.1, 0.01, …

Here we will take the numbers 0.1 and 0.01 .

Divide the second term by the first term to find the common ratio, r .

The next three terms in the sequence are 0.001, 0.0001, and 0.00001.

Example 4: continuing a geometric sequence involving fractions

Calculate the next three terms for the sequence

Here we will take the numbers 5 and 2\frac{1}{2} .

The next three terms are

\frac{5}{16}, \frac{5}{32}, and \frac{5}{64} .

Geometric sequence practice questions – continue the sequence

1. Write the next three terms of the sequence 0.5, 5, 50, 500, …

Choose two consecutive terms. For example, 5 and 50 .

Common ratio,

2. Write the next three terms of the sequence 0.04, 0.2, 1, 5, 25, …

Choose two consecutive terms. For example, 5 and 25 .

3. Calculate the next 3 terms of the sequence -1, -3, -9, -27, -81, …

Choose two consecutive terms. For example, -27 and -9 .

4. By finding the common ratio, state the next 3 terms of the sequence 640, 160, 40, 10, 2.5 .

Choose two consecutive terms. For example, 40 and 10 .

5. Work out the common ratio and therefore the next three terms in the sequence 36, 12, 4, \frac{4}{3}, \frac{4}{9}, …

Choose two consecutive terms. For example, 12 and 4 .

6. Find the common ratio and hence calculate the next three terms of the sequence 1, -1, 1, -1, 1, …

Choose two consecutive terms. For example, -1 and 1 .

How to find missing numbers in a geometric sequence

The common ratio can be used to find missing numbers in a geometric sequence. To find missing numbers in a geometric sequence you need to:

Calculate the common ratio between two consecutive terms.

Multiply the term before any missing value by the common ratio.

Divide the term after any missing value by the common ratio.

Repeat Steps 2 and 3 until all missing values are calculated. You may only need to use Step 2 or 3 depending on what terms you have been given.

Explain how to find missing numbers in a geometric sequence

Finding missing numbers in a geometric sequence examples

Example 5: find the missing numbers in the geometric sequence.

Fill in the missing terms in the sequence 7, 14, …, …,112 .

Multiply the term before any missing value by the common ratio.

The missing terms are 28 and 56 .

Note: Here, you could repeat Step 2 by using 28 \times 2 = 56.

Example 6: find the missing numbers in a geometric sequence including decimals

Find the missing values in the sequence 0.4, …, ..., 137.2, 960.4.

Divide the term after any missing value by the common ratio.

The missing terms are 2.8 and 19.6 .

Example 7: find the missing numbers in a geometric sequence when there are multiple consecutive terms missing

Find the missing values in the sequence, -4, ..., …, -108,...

First, we need to find the factor between the two terms, -108 \div -4 = 27 .

To get from -108 to -4 , we jump 3 terms.

This means that -4 has been multiplied by the common ratio three times or -4 \times r \times r \times r = -4r^3 .

\begin{aligned} r^{3}&=27\\\\ r&=3 \end{aligned}

Note: Term -108 is already given.

The missing terms are -12, -36, and -324.

We don’t need to complete this step.

Example 8: find the missing numbers in a geometric sequence including mixed numbers

Find the missing values in the sequence

Calculate the common ratio between two consecutive terms.

Repeat this step to find the next term.

40 \frac{1}{2} \times 3=121 \frac{1}{2}

The missing terms in the sequence are

1 \frac{1}{2}, 40 \frac{1}{2}, and 121 \frac{1}{2} .

Geometric sequence practice questions – find missing numbers

1. Find the missing numbers in the geometric sequence 4, 2, …, 0.5, …

Choose two consecutive terms. For example, 4 and 2 .

2. Find the missing numbers in the sequence -7, -35, …, …, -4375

Choose two consecutive terms. For example, -7 and -35 .

3. Find the missing terms in the sequence 0.6, …, …, 0.075, 0.0375

Choose two consecutive terms. For example, 0.075 and 0.0375 .

4. Calculate the missing terms in the arithmetic sequence 2 \frac{1}{5}, \frac{11}{20}, \frac{11}{80}, \ldots, \ldots

Choose two consecutive terms. For example, \frac{11}{20} and \frac{11}{80} .

5. Work out the missing terms in the sequence 3, …, …, 24 .

3 has been multiplied by the common ratio, r, three times to get 24.

3 \times r \times r \times r=24 \text{ or } 3r^{3}=24 .

Solving the equation,

6. Work out the missing terms in the sequence 90, …, …, \frac{10}{3} .

90 has been multiplied by the common ratio, r, three times to get \frac{10}{3}.

90 \times r \times r \times r=\frac{10}{3} \text{ or } 90r^{3}=\frac{10}{3} .

How to generate a geometric sequence

In order to generate a geometric sequence, we need to know the n^{th} term. Using a as the first term of the sequence, r as the common ratio and n to represent the position of the term, the n^{th} term of a geometric sequence is written as ar^{n-1}.

Once we know the first term and the common ratio, we can work out any number of terms in the sequence.

The first term is found when n=1 , the second term when n=2 , the third term when n=3 and so on.

To generate a geometric sequence you need to:

Substitute n=1 into the n^{th} term to calculate the first term.
Substitute n=2 into the n^{th} term to calculate the second term.

Continue to substitute values for n until all the required terms of the sequence are calculated.

Explain how to generate a geometric sequence

Generating a geometric sequence examples

Example 9: generate a geometric sequence using the n th term.

Generate the first 5 terms of the sequence 4^{n-1} .

Substitute n = 1 into the n^{th} term to calculate the first term.

When n = 1,\quad 4^{1-1} = 4^{0} = 1 .

Substitute n = 2 into the n^{th} term to calculate the second term.

When n = 2,\quad 4^{2-1 }= 4^{1} = 4 .

When n=3, \quad 4^{3-1}=4^{2}=16 .

When n=4, \quad 4^{4-1}=4^{3}=64 .

When n=5, \quad 4^{5-1}=4^{4}=256 .

The first 5 terms of the sequence are 1, 4, 16, 64, 256.

Example 10: generate a geometric sequence using a table

Complete the table for the first 5 terms of the arithmetic sequence 6 \times 2^{n-1}.

Example 11: generate larger terms in a geometric sequence

A geometric sequence has the n^{th} term \left(\frac{1}{2}\right)^{n} .

Calculate the 1^{st}, 2^{nd}, 10^{th} and 12^{th} terms in the sequence. Express your answers as fractions.

When n=1,\quad \left(\frac{1}{2}\right)^{1}=\frac{1}{2} .

Substitute n = 2 into the n^{th} term to calculate the second term.

When n=2, \quad \left(\frac{1}{2}\right)^{2}=\frac{1}{4} .

When n=10,\quad \left(\frac{1}{2}\right)^{10}=\frac{1}{1024} .

When n=12,\quad \left(\frac{1}{2}\right)^{12}=\frac{1}{4096} .

The unknown terms are

1, \frac{1}{4}, \frac{1}{1024}, and \frac{1}{4096} .

Example 12: generate a geometric sequence with a negative common ratio

Generate the first 5 terms of the geometric sequence 2(- 3)^{n-1} .

When n=1, \quad 2(-3)^{n-1}=2(-3)^{1-1}=2(-3)^{0}=2 \times 1=2 .

When n=2,\quad 2(-3)^{n-1}=2(-3)^{2-1}=2(-3)^{1}=2 \times-3=-6 .

When n=3, \quad 2(-3)^{n-1}=2(-3)^{3-1}=2(-3)^{2}=2 \times 9=18 .

When n=4, \quad 2(-3)^{n-1}=2(-3)^{4-1}=2(-3)^{3}=2 \times-27=-54 .

When n=5, \quad 2(-3)^{n-1}=2(-3)^{5-1}=2(-3)^{4}=2 \times 81=162 .

The first 5 terms of the sequence are 2, -6, 18, -54, and 162 .

Geometric sequence practice questions – generate a sequence

1. Generate the first 5 terms of the sequence 10^{n} .

When n=1, 10^{1}=10 .

When n=2, 10^{2}=100 .

When n=3, 10^{3}=1000 .

When n=4, 10^{4}=10000 .

When n=5, 10^{5}=100000 .

2. Generate the first 5 terms of the sequence 5^{n-1} .

When n=1, 5^{1-1}=5^{0}=1 .

When n=2, 5^{2-1}=5^{1}=5 .

When n=3, 5^{3-1}=5^{2}=25 .

When n=4, 5^{4-1}=5^{3}=125 .

When n=5, 5^{5-1}=5^{4}=625 .

3. Generate the first 5 terms of the sequence 4 \times 3^{n-1} .

When n=1, 4 \times 3^{1-1}=4 \times 3^{0}=4 .

When n=2, 4 \times 3^{2-1}=4 \times 3^{1}=12 .

When n=3, 4 \times 3^{3-1}=4 \times 3^{2}=36 .

When n=4, 4 \times 3^{4-1}=4 \times 3^{3}=108 .

When n=5, 4 \times 3^{5-1}=4 \times 3^{4}=324 .

4. Generate the first 5 terms of the sequence \frac{3^{n}}{6} .

When n=1, \frac{3^1}{6}= \frac{1}{2} .

When n=2, \frac{3^2}{6}= \frac{9}{6} = 1 \frac{1}{2} .

When n=3, \frac{3^3}{6}= \frac{27}{6} = 4 \frac{1}{2} .

When n=4, \frac{3^4}{6}= \frac{81}{6} = 13 \frac{1}{2} .

When n=5, \frac{3^5}{6}= \frac{243}{6} = 40 \frac{1}{2} .

5. Calculate the 1st, 3rd, 10th and 15th term of the sequence 2^{n} .

When n=1, 2^{1}= 2 .

When n=3, 2^{3}= 8 .

When n=10, 2^{10}= 1024 .

When n=15, 2^{15}= 32768 .

6. Calculate the first 5 terms of the sequence 3 \times (-5)^{n-1} .

When n=1, 3 \times (-5)^{1-1} = 3 \times (-5)^{0} = 3 .

When n=2, 3 \times (-5)^{2-1} = 3 \times (-5)^{1} = -15 .

When n=3, 3 \times (-5)^{3-1} = 3 \times (-5)^{2} = 75 .

When n=4, 3 \times (-5)^{4-1} = 3 \times (-5)^{3} = -375 .

When n=5, 3 \times (-5)^{5-1} = 3 \times (-5)^{4} = 1875 .

Geometric sequences GCSE exam questions

1. Which sequence is a geometric progression?

1, 3, 5, 7, 9,…. \quad \quad \quad 1, 3, 9, 27, 81, …..

1, 3, 6, 10, 15, …. \quad \quad 1, 0.6, 0.2, -0.2, -0.6,….

1, 3, 9, 27, 81, …..

2. Here is a geometric progression,

1, -5, 25, …., 625, …

(a) Find the common ratio.

(b) Work out the fourth term of the sequence.

25 \div -5 = – 5

Common ratio = -5

25 \times -5

3. A scientist is studying a type of bacteria. The number of bacteria over the first four days is shown below.

How many bacteria will there be on day 7?

180 \div 60 = 3

1620 \times 3 \times 3 \times 3

Common misconceptions

Mixing up the common ratio with the common difference for arithmetic sequences

Although these two phrases are similar, each successive term in a geometric sequence of numbers is calculated by multiplying the previous term by a common ratio and not by adding a common difference.

A negative value for r means that all terms in the sequence are negative

This is not always the case as when r is raised to an even power, the solution is always positive.

The first term in a geometric sequence

The first term is a . With ar^{n-1} , the first term would occur when n = 1 and so the power of r would be equal to 0 . Anything to the power of 0 is equal to 1 , leaving a as the first term in the sequence. This is usually mistaken when a = 1 as it is not clearly noted in the question for example, 2^{n-1} is the same as 1 \times 2^{n-1} .

Incorrect simplifying of the n th term

For example, 6 \times 3^{n-1} is incorrectly simplified to 18^{n-1} as 6 \times 3 = 18 .

The difference between an arithmetic and a geometric sequence

Arithmetic sequences are formed by adding or subtracting the same number. Geometric sequences are formed by multiplying or dividing the same number.

Learning checklist

You have now learned how to:

Recognise geometric sequences

The next lessons are

Inequalities
Functions in algebra
Laws of indices

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12.3 Geometric Sequences and Series

Learning objectives.

By the end of this section, you will be able to:

Determine if a sequence is geometric
Find the general term (nth term) of a geometric sequence
Find the sum of the first n n terms of a geometric sequence
Find the sum of an infinite geometric series
Apply geometric sequences and series in the real world

Be Prepared 12.7

Before you get started, take this readiness quiz.

Simplify: 24 32 . 24 32 . If you missed this problem, review Example 1.24 .

Be Prepared 12.8

Evaluate: ⓐ 3 4 3 4 ⓑ ( 1 2 ) 4 . ( 1 2 ) 4 . If you missed this problem, review Example 1.19 .

Be Prepared 12.9

If f ( x ) = 4 · 3 x , f ( x ) = 4 · 3 x , find ⓐ f ( 1 ) f ( 1 ) ⓑ f ( 2 ) f ( 2 ) ⓒ f ( 3 ) . f ( 3 ) . If you missed this problem, review Example 3.49 .

Determine if a Sequence is Geometric

We are now ready to look at the second special type of sequence, the geometric sequence.

A sequence is called a geometric sequence if the ratio between consecutive terms is always the same. The ratio between consecutive terms in a geometric sequence is r , the common ratio , where n is greater than or equal to two.

Geometric Sequence

A geometric sequence is a sequence where the ratio between consecutive terms is always the same.

The ratio between consecutive terms, a n a n − 1 , a n a n − 1 , is r , the common ratio . n is greater than or equal to two.

Consider these sequences.

Example 12.21

Determine if each sequence is geometric. If so, indicate the common ratio.

ⓐ 4 , 8 , 16 , 32 , 64 , 128 , … 4 , 8 , 16 , 32 , 64 , 128 , …

ⓑ −2 , 6 , −12 , 36 , −72 , 216 , … −2 , 6 , −12 , 36 , −72 , 216 , …

To determine if the sequence is geometric, we find the ratio of the consecutive terms shown.

Try It 12.41

Determine if each sequence is geometric. If so indicate the common ratio.

ⓐ 7 , 21 , 63 , 189 , 567 , 1,701 , … 7 , 21 , 63 , 189 , 567 , 1,701 , …

ⓑ 64 , 16 , 4 , 1 , 1 4 , 1 16 , … 64 , 16 , 4 , 1 , 1 4 , 1 16 , …

Try It 12.42

ⓐ −150 , −30 , −15 , −5 , − 5 2 , 0 , … −150 , −30 , −15 , −5 , − 5 2 , 0 , …

ⓑ 5 , 10 , 20 , 40 , 80 , 160 , … 5 , 10 , 20 , 40 , 80 , 160 , …

If we know the first term, a 1 , a 1 , and the common ratio, r , we can list a finite number of terms of the sequence.

Example 12.22

Write the first five terms of the sequence where the first term is 3 and the common ratio is r = −2 . r = −2 .

We start with the first term and multiply it by the common ratio. Then we multiply that result by the common ratio to get the next term, and so on.

The sequence is 3 , −6 , 12 , −24 , 48 , … 3 , −6 , 12 , −24 , 48 , …

Try It 12.43

Write the first five terms of the sequence where the first term is 7 and the common ratio is r = −3 . r = −3 .

Try It 12.44

Write the first five terms of the sequence where the first term is 6 and the common ratio is r = −4 . r = −4 .

Find the General Term ( n th Term) of a Geometric Sequence

Just as we found a formula for the general term of a sequence and an arithmetic sequence, we can also find a formula for the general term of a geometric sequence.

Let’s write the first few terms of the sequence where the first term is a 1 a 1 and the common ratio is r . We will then look for a pattern.

As we look for a pattern in the five terms above, we see that each of the terms starts with a 1 . a 1 .

The first term, a 1 , a 1 , is not multiplied by any r . In the second term, the a 1 a 1 is multiplied by r . In the third term, the a 1 a 1 is multiplied by r two times ( r · r r · r or r 2 r 2 ). In the fourth term, the a 1 a 1 is multiplied by r three times ( r · r · r r · r · r or r 3 r 3 ) and in the fifth term, the a 1 a 1 is multiplied by r four times. In each term, the number of times a 1 a 1 is multiplied by r is one less than the number of the term. This leads us to the following

General Term ( n th term) of a Geometric Sequence

The general term of a geometric sequence with first term a 1 a 1 and the common ratio r is

We will use this formula in the next example to find the fourteenth term of a sequence.

Example 12.23

Find the fourteenth term of a sequence where the first term is 64 and the common ratio is r = 1 2 . r = 1 2 .

Try It 12.45

Find the thirteenth term of a sequence where the first term is 81 and the common ratio is r = 1 3 . r = 1 3 .

Try It 12.46

Find the twelfth term of a sequence where the first term is 256 and the common ratio is r = 1 4 . r = 1 4 .

Sometimes we do not know the common ratio and we must use the given information to find it before we find the requested term.

Example 12.24

Find the twelfth term of the sequence 3, 6, 12, 24, 48, 96, … Find the general term for the sequence.

To find the twelfth term, we use the formula, a n = a 1 r n − 1 , a n = a 1 r n − 1 , and so we need to first determine a 1 a 1 and the common ratio r .

Try It 12.47

Find the ninth term of the sequence 6, 18, 54, 162, 486, 1,458, … Then find the general term for the sequence.

Try It 12.48

Find the eleventh term of the sequence 7, 14, 28, 56, 112, 224, … Then find the general term for the sequence.

Find the Sum of the First n Terms of a Geometric Sequence

We found the sum of both general sequences and arithmetic sequence. We will now do the same for geometric sequences. The sum, S n , S n , of the first n terms of a geometric sequence is written as S n = a 1 + a 2 + a 3 + ... + a n . S n = a 1 + a 2 + a 3 + ... + a n . We can write this sum by starting with the first term, a 1 , a 1 , and keep multiplying by r to get the next term as:

Let’s also multiply both sides of the equation by r .

Next, we subtract these equations. We will see that when we subtract, all but the first term of the top equation and the last term of the bottom equation subtract to zero.

Sum of the First n Terms of a Geometric Series

The sum, S n , S n , of the first n terms of a geometric sequence is

where a 1 a 1 is the first term and r is the common ratio, and r is not equal to one.

We apply this formula in the next example where the first few terms of the sequence are given. Notice the sum of a geometric sequence typically gets very large when the common ratio is greater than one.

Example 12.25

Find the sum of the first 20 terms of the geometric sequence 7, 14, 28, 56, 112, 224, …

To find the sum, we will use the formula S n = a 1 ( 1 − r n ) 1 − r . S n = a 1 ( 1 − r n ) 1 − r . We know a 1 = 7 , a 1 = 7 , r = 2 , r = 2 , and n = 20 . n = 20 .

Try It 12.49

Find the sum of the first 20 terms of the geometric sequence 3, 6, 12, 24, 48, 96, …

Try It 12.50

Find the sum of the first 20 terms of the geometric sequence 6, 18, 54, 162, 486, 1,458, …

In the next example, we are given the sum in summation notation. While adding all the terms might be possible, most often it is easiest to use the formula to find the sum of the first n terms.

To use the formula, we need r . We can find it by writing out the first few terms of the sequence and find their ratio. Another option is to realize that in summation notation, a sequence is written in the form ∑ i = 1 k a ( r ) i , ∑ i = 1 k a ( r ) i , where r is the common ratio.

Example 12.26

Find the sum: ∑ i = 1 15 2 ( 3 ) i . ∑ i = 1 15 2 ( 3 ) i .

To find the sum, we will use the formula S n = a 1 ( 1 − r n ) 1 − r , S n = a 1 ( 1 − r n ) 1 − r , which requires a 1 a 1 and r . We will write out a few of the terms, so we can get the needed information.

Try It 12.51

Find the sum: ∑ i = 1 15 6 ( 2 ) i . ∑ i = 1 15 6 ( 2 ) i .

Try It 12.52

Find the sum: ∑ i = 1 10 5 ( 2 ) i . ∑ i = 1 10 5 ( 2 ) i .

Find the Sum of an Infinite Geometric Series

If we take a geometric sequence and add the terms, we have a sum that is called a geometric series. An infinite geometric series is an infinite sum whose first term is a 1 a 1 and common ratio is r and is written

Infinite Geometric Series

An infinite geometric series is an infinite sum whose first term is a 1 a 1 and common ratio is r and is written

We know how to find the sum of the first n terms of a geometric series using the formula, S n = a 1 ( 1 − r n ) 1 − r . S n = a 1 ( 1 − r n ) 1 − r . But how do we find the sum of an infinite sum?

Let’s look at the infinite geometric series 3 + 6 + 12 + 24 + 48 + 96 + … . 3 + 6 + 12 + 24 + 48 + 96 + … . Each term gets larger and larger so it makes sense that the sum of the infinite number of terms gets larger. Let’s look at a few partial sums for this series. We see a 1 = 3 a 1 = 3 and r = 2 r = 2

As n gets larger and larger, the sum gets larger and larger. This is true when | r | ≥ 1 | r | ≥ 1 and we call the series divergent. We cannot find a sum of an infinite geometric series when | r | ≥ 1 . | r | ≥ 1 .

Let’s look at an infinite geometric series whose common ratio is a fraction less than one, 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + … 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + … . Here the terms get smaller and smaller as n gets larger. Let’s look at a few finite sums for this series. We see a 1 = 1 2 a 1 = 1 2 and r = 1 2 . r = 1 2 .

Notice the sum gets larger and larger but also gets closer and closer to one. When | r | < 1 , | r | < 1 , the expression r n r n gets smaller and smaller. In this case, we call the series convergent. As n approaches infinity, (gets infinitely large), r n r n gets closer and closer to zero. In our sum formula, we can replace the r n r n with zero and then we get a formula for the sum, S , for an infinite geometric series when | r | < 1 . | r | < 1 .

This formula gives us the sum of the infinite geometric sequence. Notice the S does not have the subscript n as in S n S n as we are not adding a finite number of terms.

Sum of an Infinite Geometric Series

For an infinite geometric series whose first term is a 1 a 1 and common ratio r ,

If | r | < 1 , the sum is If | r | < 1 , the sum is

If | r | ≥ 1 , the infinite geometric series does not have a sum. We say the series diverges. If | r | ≥ 1 , the infinite geometric series does not have a sum. We say the series diverges.

Example 12.27

Find the sum of the infinite geometric series 54 + 18 + 6 + 2 + 2 3 + 2 9 + … 54 + 18 + 6 + 2 + 2 3 + 2 9 + …

To find the sum, we first have to verify that the common ratio | r | < 1 | r | < 1 and then we can use the sum formula S = a 1 1 − r . S = a 1 1 − r .

Try It 12.53

Find the sum of the infinite geometric series 48 + 24 + 12 + 6 + 3 + 3 2 + … 48 + 24 + 12 + 6 + 3 + 3 2 + …

Try It 12.54

Find the sum of the infinite geometric series 64 + 16 + 4 + 1 + 1 4 + 1 16 + … 64 + 16 + 4 + 1 + 1 4 + 1 16 + …

An interesting use of infinite geometric series is to write a repeating decimal as a fraction.

Example 12.28

Write the repeating decimal 0. 5 – 0. 5 – as a fraction.

Try It 12.55

Write the repeating decimal 0. 4 – 0. 4 – as a fraction.

Try It 12.56

Write the repeating decimal 0. 8 – 0. 8 – as a fraction.

Apply Geometric Sequences and Series in the Real World

One application of geometric sequences has to do with consumer spending. If a tax rebate is given to each household, the effect on the economy is many times the amount of the individual rebate.

Example 12.29

The government has decided to give a $1,000 tax rebate to each household in order to stimulate the economy. The government statistics say that each household will spend 80% of the rebate in goods and services. The businesses and individuals who benefitted from that 80% will then spend 80% of what they received and so on. The result is called the multiplier effect. What is the total effect of the rebate on the economy?

Every time money goes into the economy, 80% of it is spent and is then in the economy to be spent. Again, 80% of this money is spent in the economy again. This situation continues and so leads us to an infinite geometric series.

Here the first term is 1,000, a 1 = 1000 . a 1 = 1000 . The common ratio is 0.8 , 0.8 , r = 0.8 . r = 0.8 . We can evaluate this sum since 0.8 < 1 . 0.8 < 1 . We use the formula for the sum on an infinite geometric series.

The total effect of the $1,000 received by each household will be a $5,000 growth in the economy.

Try It 12.57

What is the total effect on the economy of a government tax rebate of $1,000 to each household in order to stimulate the economy if each household will spend 90% of the rebate in goods and services?

Try It 12.58

What is the total effect on the economy of a government tax rebate of $500 to each household in order to stimulate the economy if each household will spend 85% of the rebate in goods and services?

We have looked at a compound interest formula where a principal, P , is invested at an interest rate, r , for t years. The new balance, A , is A = P ( 1 + r n ) n t A = P ( 1 + r n ) n t when interest is compounded n times a year. This formula applies when a lump sum was invested upfront and tells us the value after a certain time period.

An annuity is an investment that is a sequence of equal periodic deposits. We will be looking at annuities that pay the interest at the time of the deposits. As we develop the formula for the value of an annuity, we are going to let n = 1 . n = 1 . That means there is one deposit per year.

Suppose P dollars is invested at the end of each year. One year later that deposit is worth P ( 1 + r ) 1 P ( 1 + r ) 1 dollars, and another year later it is worth P ( 1 + r ) 2 P ( 1 + r ) 2 dollars. After t years, it will be worth A = P ( 1 + r ) t A = P ( 1 + r ) t dollars.

After three years, the value of the annuity is

This a sum of the terms of a geometric sequence where the first term is P and the common ratio is 1 + r . 1 + r . We substitute these values into the sum formula. Be careful, we have two different uses of r . The r in the sum formula is the common ratio of the sequence. In this case, that is 1 + r 1 + r where r is the interest rate.

Remember our premise was that one deposit was made at the end of each year.

We can adapt this formula for n deposits made per year and the interest is compounded n times a year.

Value of an Annuity with Interest Compounded n n Times a Year

For a principal, P , invested at the end of a compounding period, with an interest rate, r , which is compounded n times a year, the new balance, A, after t years, is

Example 12.30

New parents decide to invest $100 per month in an annuity for their baby daughter. The account will pay 5% interest per year which is compounded monthly. How much will be in the child’s account at her eighteenth birthday?

To find the Annuity formula, A t = P ( ( 1 + r n ) n t − 1 ) r n , A t = P ( ( 1 + r n ) n t − 1 ) r n , we need to identify P , r , n , and t .

Try It 12.59

New grandparents decide to invest $200 per month in an annuity for their grandson. The account will pay 5% interest per year which is compounded monthly. How much will be in the child’s account at his twenty-first birthday?

Try It 12.60

Arturo just got his first full-time job after graduating from college at age 27. He decided to invest $200 per month in an IRA (an annuity). The interest on the annuity is 8%, which is compounded monthly. How much will be in the Arturo’s account when he retires at his sixty-seventh birthday?

Access these online resources for additional instruction and practice with sequences.

Geometric Sequences
Geometric Series
Future Value Annuities and Geometric Series
Application of a Geometric Series: Tax Rebate

Section 12.3 Exercises

Practice makes perfect.

In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.

3 , 12 , 48 , 192 , 768 , 3072 , … 3 , 12 , 48 , 192 , 768 , 3072 , …

2 , 10 , 50 , 250 , 1250 , 6250 , … 2 , 10 , 50 , 250 , 1250 , 6250 , …

72 , 36 , 18 , 9 , 9 2 , 9 4 , … 72 , 36 , 18 , 9 , 9 2 , 9 4 , …

54 , 18 , 6 , 2 , 2 3 , 2 9 , … 54 , 18 , 6 , 2 , 2 3 , 2 9 , …

−3 , 6 , −12 , 24 , −48 , 96 , … −3 , 6 , −12 , 24 , −48 , 96 , …

2 , −6 , 18 , −54 , 162 , −486 , … 2 , −6 , 18 , −54 , 162 , −486 , …

In the following exercises, determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.

48 , 24 , 12 , 6 , 3 , 3 2 , … 48 , 24 , 12 , 6 , 3 , 3 2 , …

12 , 6 , 0 , −6 , −12 , −18 , … 12 , 6 , 0 , −6 , −12 , −18 , …

−7 , −2 , 3 , 8 , 13 , 18 , … −7 , −2 , 3 , 8 , 13 , 18 , …

5 , 9 , 13 , 17 , 21 , 25 , … 5 , 9 , 13 , 17 , 21 , 25 , …

1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 1 64 , … 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 1 64 , …

4 , 8 , 12 , 24 , 48 , 96 , … 4 , 8 , 12 , 24 , 48 , 96 , …

In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.

a 1 = 4 a 1 = 4 and r = 3 r = 3

a 1 = 9 a 1 = 9 and r = 2 r = 2

a 1 = −4 a 1 = −4 and r = −2 r = −2

a 1 = −5 a 1 = −5 and r = −3 r = −3

a 1 = 27 a 1 = 27 and r = 1 3 r = 1 3

a 1 = 64 a 1 = 64 and r = 1 4 r = 1 4

In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.

Find a 11 a 11 given a 1 = 8 a 1 = 8 and r = 3 . r = 3 .

Find a 13 a 13 given a 1 = 7 a 1 = 7 and r = 2 . r = 2 .

Find a 10 a 10 given a 1 = −6 a 1 = −6 and r = −2 . r = −2 .

Find a 15 a 15 given a 1 = −4 a 1 = −4 and r = −3 . r = −3 .

Find a 10 a 10 given a 1 = 100,000 a 1 = 100,000 and r = 0.1 . r = 0.1 .

Find a 8 a 8 given a 1 = 1,000,000 a 1 = 1,000,000 and r = 0.01 . r = 0.01 .

In the following exercises, find the indicated term of the given sequence. Find the general term for the sequence.

Find a 9 a 9 of the sequence, 9 , 18 , 36 , 72 , 144 , 288 , … 9 , 18 , 36 , 72 , 144 , 288 , …

Find a 12 a 12 of the sequence, 5 , 15 , 45 , 135 , 405 , 1215 , … 5 , 15 , 45 , 135 , 405 , 1215 , …

Find a 15 a 15 of the sequence, −486 , 162 , −54 , 18 , −6 , 2 , … −486 , 162 , −54 , 18 , −6 , 2 , …

Find a 16 a 16 of the sequence, 224 , −112 , 56 , −28 , 14 , −7 , … 224 , −112 , 56 , −28 , 14 , −7 , …

Find a 10 a 10 of the sequence, 1 , 0.1 , 0.01 , 0.001 , 0.0001 , 0.00001 , … 1 , 0.1 , 0.01 , 0.001 , 0.0001 , 0.00001 , …

Find a 9 a 9 of the sequence, 1000 , 100 , 10 , 1 , 0.1 , 0.01 , … 1000 , 100 , 10 , 1 , 0.1 , 0.01 , …

Find the Sum of the First n terms of a Geometric Sequence

In the following exercises, find the sum of the first fifteen terms of each geometric sequence.

8 , 24 , 72 , 216 , 648 , 1944 , … 8 , 24 , 72 , 216 , 648 , 1944 , …

7 , 14 , 28 , 56 , 112 , 224 , … 7 , 14 , 28 , 56 , 112 , 224 , …

−6 , 12 , −24 , 48 , −96 , 192 , … −6 , 12 , −24 , 48 , −96 , 192 , …

−4 , 12 , −36 , 108 , −324 , 972 , … −4 , 12 , −36 , 108 , −324 , 972 , …

81 , 27 , 9 , 3 , 1 , 1 3 , … 81 , 27 , 9 , 3 , 1 , 1 3 , …

256 , 64 , 16 , 4 , 1 , 1 4 , 1 16 , … 256 , 64 , 16 , 4 , 1 , 1 4 , 1 16 , …

In the following exercises, find the sum of the geometric sequence.

∑ i = 1 15 ( 2 ) i ∑ i = 1 15 ( 2 ) i

∑ i = 1 10 ( 3 ) i ∑ i = 1 10 ( 3 ) i

∑ i = 1 9 4 ( 2 ) i ∑ i = 1 9 4 ( 2 ) i

∑ i = 1 8 5 ( 3 ) i ∑ i = 1 8 5 ( 3 ) i

∑ i = 1 10 9 ( 1 3 ) i ∑ i = 1 10 9 ( 1 3 ) i

∑ i = 1 15 4 ( 1 2 ) i ∑ i = 1 15 4 ( 1 2 ) i

In the following exercises, find the sum of each infinite geometric series.

1 + 1 3 + 1 9 + 1 27 + 1 81 + 1 243 + 1 729 + … 1 + 1 3 + 1 9 + 1 27 + 1 81 + 1 243 + 1 729 + …

1 + 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + … 1 + 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + 1 64 + …

6 − 2 + 2 3 − 2 9 + 2 27 − 2 81 + … 6 − 2 + 2 3 − 2 9 + 2 27 − 2 81 + …

−4 + 2 − 1 + 1 2 − 1 4 + 1 8 − … −4 + 2 − 1 + 1 2 − 1 4 + 1 8 − …

6 + 12 + 24 + 48 + 96 + 192 + … 6 + 12 + 24 + 48 + 96 + 192 + …

5 + 15 + 45 + 135 + 405 + 1215 + … 5 + 15 + 45 + 135 + 405 + 1215 + …

1,024 + 512 + 256 + 128 + 64 + 32 + … 1,024 + 512 + 256 + 128 + 64 + 32 + …

6,561 + 2187 + 729 + 243 + 81 + 27 + … 6,561 + 2187 + 729 + 243 + 81 + 27 + …

In the following exercises, write each repeating decimal as a fraction.

0. 3 – 0. 3 –

0. 6 – 0. 6 –

0. 7 – 0. 7 –

0. 2 – 0. 2 –

0. 45 — 0. 45 —

0. 27 — 0. 27 —

In the following exercises, solve the problem.

Find the total effect on the economy of each government tax rebate to each household in order to stimulate the economy if each household will spend the indicated percent of the rebate in goods and services.

New grandparents decide to invest $ 100 $ 100 per month in an annuity for their grandchild. The account will pay 6 % 6 % interest per year which is compounded monthly (12 times a year). How much will be in the child’s account at their twenty-first birthday?

Berenice just got her first full-time job after graduating from college at age 30. She decided to invest $ 500 $ 500 per quarter in an IRA (an annuity). The interest on the annuity is 7 % 7 % which is compounded quarterly (4 times a year). How much will be in the Berenice’s account when she retires at age 65?

Alice wants to purchase a home in about five years. She is depositing $ 500 $ 500 a month into an annuity that earns 5 % 5 % per year that is compounded monthly (12 times a year). How much will Alice have for her down payment in five years?

Myra just got her first full-time job after graduating from college. She plans to get a master’s degree, and so is depositing $ 2,500 $ 2,500 a year from her year-end bonus into an annuity. The annuity pays 6.5 % 6.5 % per year and is compounded yearly. How much will she have saved in five years to pursue her master’s degree?

Writing Exercises

In your own words, explain how to determine whether a sequence is geometric.

In your own words, explain how to find the general term of a geometric sequence.

In your own words, explain the difference between a geometric sequence and a geometric series.

In your own words, explain how to determine if an infinite geometric series has a sum and how to find it.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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Authors: Lynn Marecek, Andrea Honeycutt Mathis
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Book title: Intermediate Algebra 2e
Publication date: May 6, 2020
Location: Houston, Texas
Book URL: https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
Section URL: https://openstax.org/books/intermediate-algebra-2e/pages/12-3-geometric-sequences-and-series

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Geometric Sequences and Sums

A Sequence is a set of things (usually numbers) that are in order.

Geometric Sequences

In a Geometric Sequence each term is found by multiplying the previous term by a constant .

Example: 1, 2, 4, 8, 16, 32, 64, 128, 256, ...

This sequence has a factor of 2 between each number.

Each term (except the first term) is found by multiplying the previous term by 2 .

In General we write a Geometric Sequence like this:

{a, ar, ar 2 , ar 3 , ... }

a is the first term, and
r is the factor between the terms (called the "common ratio" )

Example: {1,2,4,8,...}

The sequence starts at 1 and doubles each time, so

a=1 (the first term)
r=2 (the "common ratio" between terms is a doubling)

And we get:

= {1, 1×2, 1×2 2 , 1×2 3 , ... }

= {1, 2, 4, 8, ... }

But be careful, r should not be 0:

When r=0 , we get the sequence {a,0,0,...} which is not geometric

We can also calculate any term using the Rule:

x n = ar (n-1)

(We use "n-1" because ar 0 is for the 1st term)

Example: 10, 30, 90, 270, 810, 2430, ...

This sequence has a factor of 3 between each number.

The values of a and r are:

a = 10 (the first term)
r = 3 (the "common ratio")

The Rule for any term is:

x n = 10 × 3 (n-1)

So, the 4th term is:

x 4 = 10 × 3 (4-1) = 10 × 3 3 = 10 × 27 = 270

And the 10th term is:

x 10 = 10 × 3 (10-1) = 10 × 3 9 = 10 × 19683 = 196830

A Geometric Sequence can also have smaller and smaller values:

Example: 4, 2, 1, 0.5, 0.25, ...

This sequence has a factor of 0.5 (a half) between each number.

Its Rule is x n = 4 × (0.5) n-1

Why "Geometric" Sequence?

Because it is like increasing the dimensions in geometry :

Geometric Sequences are sometimes called Geometric Progressions (G.P.’s)

Summing a Geometric Series

To sum these:

a + ar + ar 2 + ... + ar (n-1)

(Each term is ar k , where k starts at 0 and goes up to n-1)

We can use this handy formula:

What is that funny Σ symbol? It is called Sigma Notation

And below and above it are shown the starting and ending values:

It says "Sum up n where n goes from 1 to 4. Answer= 10

The formula is easy to use ... just "plug in" the values of a , r and n

Example: Sum the first 4 terms of 10, 30, 90, 270, 810, 2430, ...

The values of a , r and n are:

n = 4 (we want to sum the first 4 terms)

You can check it yourself:

10 + 30 + 90 + 270 = 400

And, yes, it is easier to just add them in this example , as there are only 4 terms.

But imagine adding 50 terms ... then the formula is much easier.

Using the Formula

Let's see the formula in action:

Example: Grains of Rice on a Chess Board

On the page Binary Digits we give an example of grains of rice on a chess board. The question is asked:

When we place rice on a chess board:

1 grain on the first square,
2 grains on the second square,
4 grains on the third and so on,

... doubling the grains of rice on each square ...

... how many grains of rice in total?

So we have:

a = 1 (the first term)
r = 2 (doubles each time)
n = 64 (64 squares on a chess board)

= 1−2 64 −1 = 2 64 − 1

= 18,446,744,073,709,551,615

Which was exactly the result we got on the Binary Digits page (thank goodness!)

And another example, this time with r less than 1:

Example: Add up the first 10 terms of the Geometric Sequence that halves each time:

{ 1/2, 1/4, 1/8, 1/16, ... }.

a = ½ (the first term)
r = ½ (halves each time)
n = 10 (10 terms to add)

Very close to 1.

(Question: if we continue to increase n , what happens?)

Why Does the Formula Work?

Let's see why the formula works, because we get to use an interesting "trick" which is worth knowing.

Notice that S and S·r are similar?

Now subtract them!

Wow! All the terms in the middle neatly cancel out. (Which is a neat trick)

By subtracting S·r from S we get a simple result:

S − S·r = a − ar n

Let's rearrange it to find S :

Which is our formula (ta-da!):

Infinite Geometric Series

So what happens when n goes to infinity ?

We can use this formula:

But be careful :

r must be between (but not including) −1 and 1

and r should not be 0 because the sequence {a,0,0,...} is not geometric

So our infnite geometric series has a finite sum when the ratio is less than 1 (and greater than −1)

Let's bring back our previous example, and see what happens:

Example: Add up ALL the terms of the Geometric Sequence that halves each time:

{ 1 2 , 1 4 , 1 8 , 1 16 , ... }.

= ½×1 ½ = 1

Yes, adding 1 2 + 1 4 + 1 8 + ... etc equals exactly 1 .

Recurring Decimal

On another page we asked "Does 0.999... equal 1?" , well, let us see if we can calculate it:

Example: Calculate 0.999...

We can write a recurring decimal as a sum like this:

And now we can use the formula:

Yes! 0.999... does equal 1.

So there we have it ... Geometric Sequences (and their sums) can do all sorts of amazing and powerful things.

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Geometric sequence word problems

This lesson will show you how to solve a variety of geometric sequence word problems.

Example #1:

The stock's price of a company is not doing well lately. Suppose the stock's price is 92% of its previous price each day. What is the stock's price after 10 days if the stock was worth $2500 right before it started to go down?

To solve this problem, we need the geometric sequence formula shown below.

a n = a 1 × r (n - 1)

a 1 = original value of the stock = 2500

a 2 = value of the stock after 1 day

a 11 = value of the stock after 10 days

a 11 = 2500 × (0.92) (11 - 1)

a 11 = 2500 × (0.92) 10

a 11 = 2500 × 0.434

a 11 = $1085

The stock's price is about 1085 dollars.

Example #2:

The third term of a geometric sequence is 45 and the fifth term of the geometric sequence is 405. If all the terms of the sequence are positive numbers, find the 15th term of the geometric sequence.

Solution To solve this problem, we need the geometric sequence formula shown below.

a n = a 1 × r (n - 1)

Find the third term

a 3 = a 1 × r (3 - 1)

a 3 = a 1 × r 2

Since the third term is 45, 45 = a 1 × r 2 ( equation 1 )

Find the fifth term

a 5 = a 1 × r (5 - 1)

a 5 = a 1 × r 4

Since the fifth term is 405, 405 = a 1 × r 4 ( equation 2 )

Divide equation 2 by equation 1 .

(a 1 × r 4 ) / (a 1 × r 2 ) = 405 / 45

Cancel a 1 since it is both on top and at the bottom of the fraction.

r 4 / r 2 = 9

r = ±√9

r = ±3

Use r = 3, and equation 1 to find a 1

45 = a 1 × (3) 2

45 = a 1 × 9

a 1 = 45 / 9 = 5

Since all the terms of the sequence are positive numbers, we must use r = 3 if we want all the terms to be positive numbers.

Let us now find a 15

a 15 = 5 × (3) (15 - 1)

a 15 = 5 × (3) 14

a 15 = 5 × 4782969

a 15 = 23914845

Challenging geometric sequence word problems

Example #3:

Suppose that the magnification of a PDF file on a desktop computer is increased by 15% for each level of zoom. Suppose also that the original length of the word " January " is 1.2 cm. Find the length of the word " January " after 6 magnifications.

a 1 = original length of the word = 1.2 cm

a 2 = length of the word after 1 magnification

a 7 = length of the word after 6 magnifications

r = 1 + 0.15 = 1.15

a 7 = 1.2 × (1.15) (7 - 1)

a 7 = 1.2 × (1.15) 6

a 7 = 1.2 × 2.313

a 7 = 2.7756

After 6 magnifications, the length of the word "January" is 2.7756 cm.

Notice that we added 1 to 0.15. Why did we do that? Let us not use the formula directly so you can see the reason behind it. Study the following carefully !

Day 1 : a 1 = 1.2

Day 2 : a 2 = 1.2 + 1.2 (0.15) = 1.2 (1 + 0.15)

Day 3 : a 3 = 1.2(1 + 0.15) + [ 1.2(1 + 0.15) ]0.15 = 1.2(1 + 0.15) (1 + 0.15) = 1.2(1 + 0.15) 2

Day 7 : a 7 = 1.2(1 + 0.15) 6

Suppose that you want a reduced copy of a photograph. The actual length of the photograph is 10 inches. If each reduction is 64% of the original, how many reductions, will shrink the photograph to 1.07 inches.

a 1 = original length of the photograph = 10 inches

a 2 = length of the photograph after 1 reduction

n = number of reductions = ?

1.07 = 10 × (0.64) (n - 1)

Divide both sides by 10

1.07 / 10 = [10 × (0.64) (n - 1) ] / 10

0.107 = (0.64) (n - 1)

Notice that you have an exponential equation to solve. The biggest challenge then is knowing how to solve exponential equations !

Take the natural log of both sides of the equation.

ln(0.107) = ln[(0.64) (n - 1) ]

Use the power property of logarithms .

ln(0.107) = (n - 1)ln(0.64)

Divide both sides of the equation by ln(0.64)

ln(0.107) / ln(0.64) = (n - 1)ln(0.64) / ln(0.64)

n - 1 = ln(0.107) / ln(0.64)

Use a calculator to find ln(0.107) and ln(0.64)

n - 1 = -2.23492644452 \ -0.44628710262

n - 1 = 5.0078

n = 1 + 5.0078

Therefore, you will need 6 reductions.

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9.3: Geometric Sequences

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Learning Objectives

Find the common ratio for a geometric sequence.
List the terms of a geometric sequence.
Use a recursive formula for a geometric sequence.
Use an explicit formula for a geometric sequence.

Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $$26,000$. He is promised a $2\%$ cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by $102\%$. His salary will be $$26,520$ after one year; $$27,050.40$ after two years; $$27,591.41$ after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section, we will review sequences that grow in this way.

Finding Common Ratios

The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio . The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.

$A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.$

Definition: GEOMETRIC SEQUENCE

A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If $a_1$ is the initial term of a geometric sequence and $r$ is the common ratio, the sequence will be

\[\{a_1, a_1r,a_1r^2,a_1r^3,...\} .\]

How to: Given a set of numbers, determine if they represent a geometric sequence.

Divide each term by the previous term.
Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.

Example $\PageIndex{1}$: Finding Common Ratios

Is the sequence geometric? If so, find the common ratio.

$1$, $2$, $4$, $8$, $16$,...
$48$, $12$, $4$, $2$,...

Divide each term by the previous term to determine whether a common ratio exists.

The sequence is geometric because there is a common ratio. The common ratio is $2$.

The sequence is not geometric because there is not a common ratio.

The graph of each sequence is shown in Figure $\PageIndex{1}$. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.

$Graph of two sequences where graph (a) is geometric and graph (b) is exponential.$

Figure $\PageIndex{1}$

If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?

No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.

Exercise $\PageIndex{1A}$

$5$, $10$, $15$, $20$,...

The sequence is not geometric because $\dfrac{10}{5}≠\dfrac{15}{10}$

Exercise $\PageIndex{1B}$

$100$, $20$, $4$, $\dfrac{4}{5}$,...

The sequence is geometric. The common ratio is $\dfrac{1}{5}$

Writing Terms of Geometric Sequences

Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is $a_1=−2$ and the common ratio is $r=4$, we can find subsequent terms by multiplying $−2⋅4$ to get $−8$ then multiplying the result $−8⋅4$ to get $−32$ and so on.

\[\begin{align*} a_1 &= −2 \\ a_2 &= (−2⋅4)=−8 \\ a_3 &= (−8⋅4)=−32 \\ a_4 &= (−32⋅4)=128 \end{align*}\]

The first four terms are $\{–2, –8, –32, –128\}$.

How to: Given the first term and the common factor, find the first four terms of a geometric sequence.

Multiply the initial term, $a_1$, by the common ratio to find the next term, $a_2$.
Repeat the process, using $a_n=a_2$ to find $a_3$ and then use $a_3$ to find $a_4$, until all four terms have been identified.
Write the terms separated by commons within brackets.

Example $\PageIndex{2}$: Writing the Terms of a Geometric Sequence

List the first four terms of the geometric sequence with $a_1=5$ and $r=–2$.

Multiply $a_1$ by $−2$ to find $a_2$. Repeat the process, using $a_2$ to find $a_3$, and so on.

\[\begin{align*} a_1 &= 5 \\ a_2 &= −2a_1=−10 \\ a_3 &= −2a_2=20 \\ a_4 &= −2a_3=−40 \end{align*}\]

The first four terms are $\{5,–10,20,–40\}$.

Exercise $\PageIndex{2}$

List the first five terms of the geometric sequence with $a_1=18$ and $r=\dfrac{1}{3}$.

$\left \{18, 6, 2, \dfrac{2}{3}, \dfrac{2}{9} \right \} $

Using Recursive Formulas for Geometric Sequences

A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is $9$. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.

Note: RECURSIVE FORMULA FOR A GEOMETRIC SEQUENCE

The recursive formula for a geometric sequence with common ratio r and first term $a_1$ is

\[a_n=ra_{n−1},\;\;\; n≥2\]

How to: Given the first several terms of a geometric sequence, write its recursive formula.

State the initial term.
Find the common ratio by dividing any term by the preceding term.
Substitute the common ratio into the recursive formula for a geometric sequence.

Example $\PageIndex{3}$: Using Recursive Formulas for Geometric Sequences

Write a recursive formula for the following geometric sequence.

$\{6, 9, 13.5, 20.25, ...\} \nonumber$

The first term is given as $6$. The common ratio can be found by dividing the second term by the first term.

$r=\dfrac{9}{6}=1.5 \nonumber$

Substitute the common ratio into the recursive formula for geometric sequences and define $a_1$.

\[\begin{align*} a_n &= ra_{n−1} \\ a_n &= 1.5a_{n−1} \text{ for }n≥2 \\ a_1 &= 6 \end{align*}\]

The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure $\PageIndex{2}$.

$Graph of the geometric sequence.$

Figure $\PageIndex{2}$

Do we have to divide the second term by the first term to find the common ratio?

No. We can divide any term in the sequence by the previous term. It is, however, most common to divide the second term by the first term because it is often the easiest method of finding the common ratio.

Exercise $\PageIndex{3}$

$\{2, 43, 89, 1627, ...\}$

$\begin{align*}a_1 &= 2 \\ a_n &= \dfrac{2}{3}a_{n−1} \text{ for }n≥2 \end{align*}$

Using Explicit Formulas for Geometric Sequences

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.

\[a_n=a_1r^{n−1}\]

Let’s take a look at the sequence $\{18, 36, 72, 144, 288, ...\}$. This is a geometric sequence with a common ratio of $2$ and an exponential function with a base of $2$. An explicit formula for this sequence is

$a_n=18·2^{n−1}$

The graph of the sequence is shown in Figure $\PageIndex{3}$.

$Graph of the geometric sequence.$

Figure $\PageIndex{3}$

Note: EXPLICIT FORMULA FOR A GEOMETRIC SEQUENCE

The $n^{th}$ term of a geometric sequence is given by the explicit formula :

Example $\PageIndex{4}$: Writing Terms of Geometric Sequences Using the Explicit Formula

Given a geometric sequence with $a_1=3$ and $a_4=24$, find $a_2$.

The sequence can be written in terms of the initial term and the common ratio $r$.

$3$, $3r$, $3r^2$, $3r^3$,...

Find the common ratio using the given fourth term.

\[\begin{align*} a_n&=a_1r^{n-1} \\ a_4&=3r^3 \qquad \text{Write the fourth term of sequence in terms of }\alpha_1 \text{ and } r \\ 24&=3r^3 \qquad \text{Substitute }24 \text{ for }a_4 \\ 8&=r^3 \qquad \text{Divide} \\ r&=2 \qquad \text{Solve for the common ratio} \end{align*}\]

Find the second term by multiplying the first term by the common ratio.

\[\begin{align*} a_2 &= 2 \\ a_1 &= 2(3) \\ & = 6 \end{align*}\]

The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.

Exercise $\PageIndex{4}$

Given a geometric sequence with $a_2=4$ and $2a_3=32$, find $a_6$.

$a_6=16,384$

Example $\PageIndex{5}$: Writing an Explicit Formula for the n th Term of a Geometric Sequence

Write an explicit formula for the nth term of the following geometric sequence.

$\{2, 10, 50, 250, ...\}$

The first term is $2$. The common ratio can be found by dividing the second term by the first term.

$\dfrac{10}{2}=5$

The common ratio is $5$. Substitute the common ratio and the first term of the sequence into the formula.

\[\begin{align*}a_n &= a_1r^{(n−1)} \\ a_n &= 2⋅5^{n−1} \end{align*}\]

The graph of this sequence in Figure $\PageIndex{4}$ shows an exponential pattern.

$Graph of the geometric sequence.$

Figure $\PageIndex{4}$

Exercise $\PageIndex{5}$

Write an explicit formula for the following geometric sequence.

$\{–1, 3, –9, 27, ...\}$

$a_n=−{(−3)}^{n−1}$

Solving Application Problems with Geometric Sequences

In real-world scenarios involving arithmetic sequences, we may need to use an initial term of $a_0$ instead of $a_1$. In these problems, we can alter the explicit formula slightly by using the following formula:

$a_n=a_0r^n$

Example $\PageIndex{6}$: Solving Application Problems with Geometric Sequences

In 2013, the number of students in a small school is $284$. It is estimated that the student population will increase by $4\%$ each year.

Write a formula for the student population.
Estimate the student population in 2020.

Let $P$ be the student population and $n$ be the number of years after 2013. Using the explicit formula for a geometric sequence we get

$P_n =284⋅{1.04}^n$

$2020−2013=7$

We are looking for the population after $7$ years. We can substitute $7$ for $n$ to estimate the population in 2020.

$P_7=284⋅{1.04}^7≈374$

The student population will be about $374$ in 2020.

Exercise $\PageIndex{6}$

A business starts a new website. Initially the number of hits is $293$ due to the curiosity factor. The business estimates the number of hits will increase by $2.6%$ per week.

Write a formula for the number of hits.
Estimate the number of hits in $5$ weeks.

$P_n = 293⋅1.026a^n$

The number of hits will be about $333$.

Access these online resources for additional instruction and practice with geometric sequences.

Geometric Sequences
Determine the Type of Sequence
Find the Formula for a Sequence

Key Equations

Key concepts.

A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
The constant ratio between two consecutive terms is called the common ratio.
The common ratio can be found by dividing any term in the sequence by the previous term. See Example $\PageIndex{1}$.
The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See Example $\PageIndex{2}$ and Example $\PageIndex{4}$.
A recursive formula for a geometric sequence with common ratio $r$ is given by $a_n=ra_{n–1}$ for $n≥2$.
As with any recursive formula, the initial term of the sequence must be given. See Example $\PageIndex{3}$.
An explicit formula for a geometric sequence with common ratio $r$ is given by $a_n=a_1r^{n–1}$. See Example $\PageIndex{5}$.
In application problems, we sometimes alter the explicit formula slightly to $a_n=a_0r^n$. See Example $\PageIndex{6}$.

Contributors and Attributions

Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .

Geometric Sequences

Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?

Study Guides > College Algebra

Solving application problems with geometric sequences.

In real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[/latex] instead of [latex]{a}_{1}[/latex]. In these problems, we can alter the explicit formula slightly by using the following formula:

Example 7: Solving Application Problems with Geometric Sequences

Write a formula for the student population.
Estimate the student population in 2020.
The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04. Let [latex]P[/latex] be the student population and [latex]n[/latex] be the number of years after 2013. Using the explicit formula for a geometric sequence we get [latex]{P}_{n} =284\cdot {1.04}^{n}[/latex]
We can find the number of years since 2013 by subtracting. [latex]2020 - 2013=7[/latex] We are looking for the population after 7 years. We can substitute 7 for [latex]n[/latex] to estimate the population in 2020. [latex]{P}_{7}=284\cdot {1.04}^{7}\approx 374[/latex] The student population will be about 374 in 2020.

Licenses & Attributions

Cc licensed content, specific attribution.

Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/ [email protected] :1/Preface. License: CC BY: Attribution .

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Geometric Sequences and Series

Related Topics: More Lessons for Intermediate Algebra More Lessons for Algebra Math Worksheets

A series of free, online Intermediate Algebra Lessons or Algebra II lessons. Videos, worksheets, and activities to help Algebra students.

In this lesson, we will learn

geometric sequences Click here
how to find the nth term in a geometric sequence Click here
geometric series Click here
how to find the sum of a geometric series

Geometric Sequences

A list of numbers that follows a rule is called a sequence. Sequences whose rule is the multiplication of a constant are called geometric sequences, similar to arithmetic sequences that follow a rule of addition. Homework problems on geometric sequences often ask us to find the nth term of a sequence using a formula. Geometric sequences are important to understanding geometric series.

Geometric Sequences Determine the nth term of a geometric sequence. Determine the common ratio of a geometric sequence. Determine the formula for a geometric sequence.

A geometric sequence is a sequence that has the pattern of multiplying by a constant to determine the consecutive terms. We say geometric sequences have a common ratio. a n = a n - 1 r Example:

A sequence is a function. What is the domain and range of the following sequence? What is r? -12, 6, -3, 3/2, -3/4
Given the formula for geometric sequence, determine the first two terms, and then the 5th term. Also state the common ratio.
Given the geometric sequence, determine the formula, Then determine the 6th term. 1/3, 2/8, 4/17, 8/81, …

Geometric Sequences (Introduction)

A Quick Intro to Geometric Sequences Gives the definition of a geometric sequence and go through 4 examples, determining if each qualifies as a geometric sequence or not. A geometric sequence is a sequence of numbers where each term after the first term is found by multiplying the previous one by a fixed non-zero number, called the common ratio. Examples: Determine which of the following sequences are geometric. If so, give the value of the common ratio, r.

3,6,12,24,48,96, …
3, 3/2, 3/4, 3/8, 3/16, 3/32, 3/64, …
10,15,20,25,30, …
-1, .1, -.01, .001, -.0001

Geometric Sequences - Find the nth term

Geometric Sequences: A Formula for the’ n - th ’ Term. Derive the formula to find the ’n-th’ term of a geometric sequence by considering an example. The formula to find another term of the sequence. Example: Consider the geometric sequence 3,6,12,24,48,..

Derive the a n formula.

How to find the general term or nth term of a geometric sequence? Examples:

3, 3/2, 3/4, 3/8, 3/16, …
a 3 = 5, a 7 = 80

Geometric Series

We can use what we know of geometric sequences to understand geometric series. A geometric series is a series or summation that sums the terms of a geometric sequence. There are methods and formulas we can use to find the value of a geometric series. It can be helpful for understanding geometric series to understand arithmetic series, and both concepts will be used in upper-level Calculus topics.

How to determine the partial sum of a geometric series? Summing or adding the terms of a geometric sequence creates what is called a series. Example:

Determine the sum of the geometric series. 3 + 6 + 12 + … + 1536
Determine the sum of the geometric series. a n = 2(-3) n-1 , n = 5

How to find the sum of a geometric series?

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Sharpen your math proficiency with these Geometric Series Practice Problems. Solve ten (10) questions and verify your solutions by comparing them with the provided answers. ... Geometric Series Practice Problems with Answers. Once you have solved the problems on paper, click the ANSWER button to verify that you have answered the questions ...
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Geometric sequences are sequences in which the next number in the sequence is found by multiplying the previous term by a number called the common ratio. The common ratio is denoted by the letter r. Depending on the common ratio, the geometric sequence can be increasing or decreasing. If the common ratio is greater than 1, the sequence is ...
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Definition 12.4.4. An infinite geometric series is an infinite sum whose first term is a1 and common ratio is r and is written. a1 + a1r + a1r2 + … + a1rn − 1 + …. We know how to find the sum of the first n terms of a geometric series using the formula, Sn = a1(1 − rn) 1 − r.
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How to Solve Infinite Geometric Series; How to Solve Arithmetic Sequences; Step by step guide to solve Geometric Sequence Problems. It is a sequence of numbers where each term after the first is found by multiplying the previous item by the common ratio, a fixed, non-zero number. For example, the sequence $2, 4, 8, 16, 32$, … is a geometric ...
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Geometric Sequences. A geometric sequence 18, or geometric progression 19, is a sequence of numbers where each successive number is the product of the previous number and some constant $r$. \[a_{n}=r a_{n-1} \quad\color{Cerulean}{Geometric\:Sequence}\] And because $\frac{a_{n}}{a_{n-1}}=r$, the constant factor $r$ is called the common ratio 20.For example, the following is a geometric ...
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Example 1: continuing a geometric sequence. Calculate the next three terms for the geometric progression 1, 2, 4, 8, 16, 1, 2,4,8,16, …. Take two consecutive terms from the sequence. Here we will take the numbers 4 4 and 8 8. 2 Divide the second term by the first term to find the value of the common ratio, r r.
12.3 Geometric Sequences and Series
We found the sum of both general sequences and arithmetic sequence. We will now do the same for geometric sequences. The sum, Sn, of the first n terms of a geometric sequence is written as Sn = a1 + a2 + a3 + ... + an. We can write this sum by starting with the first term, a1, and keep multiplying by r to get the next term as:
Geometric sequences review (article)
Geometric sequence formulas give a ( n) , the n th term of the sequence. This is the explicit formula for the geometric sequence whose first term is k and common ratio is r : a ( n) = k ⋅ r n − 1. This is the recursive formula of that sequence: { a ( 1) = k a ( n) = a ( n − 1) ⋅ r.
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Whereas if a = 400.641 - then the 10th term would therefore be 400.641 (-0.5^9) = -0.782501953125 - which is clearly NOT the correct result. The correct answer for the first term in that geometric sequence is exactly 400. ( 2 votes) Upvote. Downvote.
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A geometric sequence is a sequence of numbers where each term after the first term is found by multiplying the previous one by a fixed non-zero number, called the common ratio. Example: Determine which of the following sequences are geometric. If so, give the value of the common ratio, r. 3,6,12,24,48,96, ….
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The geometric progression in Example 8.3.3 is a decreasing sequence. The r −value is calculated by considering the offset from 100%. If the percentage, p%, is a decrease, subtract the percentage from 100%: (100% − p%) then change the value to a decimal by moving the decimal two places left (or divide by 100 ).
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Show Video Lesson. Application of a Geometric Sequence. Example: Bouncing ball application of a geometric sequence. When a ball is dropped onto a flat floor, it bounces to 65% of the height from which it was dropped. If the ball is dropped from 80 cm, find the height of the fifth bounce. Show Video Lesson. Population Growth and Compound Interest.
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Scroll down the page for more examples and solutions for Geometric Sequences and Geometric Series. This video gives the definition of a geometric sequence and go through 4 examples, determining if each qualifies as a geometric sequence or not! Geometric Sequences: A Formula for the 'n-th' Term. This video derives the formula to find the ...
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Many real-world problems involve geometric sequences and series. The following definitions can help us solve these problems. A finite geometric sequence has the form 𝑇, 𝑇 𝑟, 𝑇 𝑟, …, 𝑇 𝑟 , where 𝑇 is the first term, 𝑟 is the common ratio, and 𝑛 is the number of terms in the sequence.
Geometric Sequences and Sums
Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents. Geometric Sequences and Sums Sequence. A Sequence is a set of things (usually numbers) that are in order. Geometric Sequences. In a Geometric Sequence each term is found by multiplying the previous term by a constant ...
Geometric sequence
A common lemma is that a sequence is in geometric progression if and only if is the geometric mean of and for any consecutive terms . In symbols, . This is mostly used to perform substitutions, though it occasionally serves as a definition of geometric sequences. Sum. A geometric series is the sum of all the terms of a geometric sequence. They ...
Geometric Sequence Word Problems
The third term of a geometric sequence is 45 and the fifth term of the geometric sequence is 405. If all the terms of the sequence are positive numbers, find the 15th term of the geometric sequence. Solution To solve this problem, we need the geometric sequence formula shown below. a n = a 1 × r (n - 1) Find the third term. a 3 = a 1 × r (3 - 1)
9.3: Geometric Sequences
Using Explicit Formulas for Geometric Sequences. Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms. an = a1rn−1 (9.3.3) (9.3.3) a n = a 1 r n − 1.
Geometric Sequences
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Solving Application Problems with Geometric Sequences
The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04. Let. P. P P be the student population and. n. n n be the number of years after 2013. Using the explicit formula for a geometric sequence we get. P n = 2 8 4 ⋅ 1. 0 4 n.
Geometric Sequences and Series
A geometric sequence is a sequence of numbers where each term after the first term is found by multiplying the previous one by a fixed non-zero number, called the common ratio. Examples: Determine which of the following sequences are geometric. If so, give the value of the common ratio, r. 3,6,12,24,48,96, ….
FGeo-DRL: Deductive Reasoning for Geometric Problems through Deep
Finally, the known conditions and objectives of the problem form the state space, while the set of theorems forms the action space. Leveraging FGeo-DRL, we have achieved readable and verifiable automated solutions to geometric problems. Experiments conducted on the formalgeo7k dataset have achieved a problem-solving success rate of 86.40%.

Geometric Sequences Problems with Solutions

Review OF Geometric Sequences

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Geometric Sequences

What is a geometric sequence?

What are geometric sequences?

Geometric sequence formula

Related lessons on sequences

How to continue a geometric sequence

Explain how to continue a geometric sequence

Geometric sequences worksheet

Continuing a geometric sequence examples

Example 2: continuing a geometric sequence with negative numbers

Example 3: continuing a geometric sequence with decimals

Example 4: continuing a geometric sequence involving fractions

Geometric sequence practice questions – continue the sequence

How to find missing numbers in a geometric sequence

Explain how to find missing numbers in a geometric sequence

Finding missing numbers in a geometric sequence examples

Example 6: find the missing numbers in a geometric sequence including decimals

Example 7: find the missing numbers in a geometric sequence when there are multiple consecutive terms missing

Example 8: find the missing numbers in a geometric sequence including mixed numbers

Geometric sequence practice questions – find missing numbers

How to generate a geometric sequence

Explain how to generate a geometric sequence

Generating a geometric sequence examples

Example 10: generate a geometric sequence using a table

Example 11: generate larger terms in a geometric sequence

Example 12: generate a geometric sequence with a negative common ratio

Geometric sequence practice questions – generate a sequence

Geometric sequences GCSE exam questions

Common misconceptions

Learning checklist

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12.3 Geometric Sequences and Series

Be Prepared 12.7

Be Prepared 12.8

Be Prepared 12.9

Geometric Sequence

Example 12.21

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Example 12.22

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General Term ( n th term) of a Geometric Sequence

Example 12.23

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Example 12.24

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Find the Sum of the First n Terms of a Geometric Sequence

Sum of the First n Terms of a Geometric Series

Example 12.25

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Example 12.26

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