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Section exercises

Verbal

What is a geometric sequence?

A sequence in which the ratio between any two consecutive terms is constant.

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How is the common ratio of a geometric sequence found?

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What is the procedure for determining whether a sequence is geometric?

Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.

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What is the difference between an arithmetic sequence and a geometric sequence?

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Describe how exponential functions and geometric sequences are similar. How are they different?

Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.

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Algebraic

For the following exercises, find the common ratio for the geometric sequence.

1 , 3 , 9 , 27 , 81 , ...

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0.125 , 0.25 , 0.5 , 1 , 2 , ...

The common ratio is 2

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2 , 1 2 , 1 8 , 1 32 , 1 128 , ...

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For the following exercises, determine whether the sequence is geometric. If so, find the common ratio.

6 , 12 , 24 , 48 , 96 , ...

The sequence is geometric. The common ratio is 2.

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5 , 5.2 , 5.4 , 5.6 , 5.8 , ...

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1 , 1 2 , 1 4 , 1 8 , 1 16 , ...

The sequence is geometric. The common ratio is 1 2 .

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6 , 8 , 11 , 15 , 20 , ...

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0.8 , 4 , 20 , 100 , 500 , ...

The sequence is geometric. The common ratio is 5.

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

a 1 = 5 , r = 1 5

5 , 1 , 1 5 , 1 25 , 1 125

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For the following exercises, write the first five terms of the geometric sequence, given any two terms.

a 6 = 25 , a 8 = 6.25

800 , 400 , 200 , 100 , 50

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For the following exercises, find the specified term for the geometric sequence, given the first term and common ratio.

The first term is 2, and the common ratio is 3. Find the 5 th term.

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The first term is 16 and the common ratio is 1 3 . Find the 4 th term.

a 4 = 16 27

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For the following exercises, find the specified term for the geometric sequence, given the first four terms.

a n = { 1 , 2 , 4 , 8 , ... } . Find a 12 .

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a n = { 2 , 2 3 , 2 9 , 2 27 , ... } . Find a 7 .

a 7 = 2 729

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For the following exercises, write the first five terms of the geometric sequence.

a 1 = 486 , a n = 1 3 a n 1

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a 1 = 7 , a n = 0.2 a n 1

7 , 1.4 , 0.28 , 0.056 , 0.0112

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For the following exercises, write a recursive formula for each geometric sequence.

a n = { 1 , 5 , 25 , 125 , ... }

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a n = { 32 , 16 , 8 , 4 , ... }

a = 1 32 , a n = 1 2 a n 1

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a n = { 14 , 56 , 224 , 896 , ... }

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a n = { 10 , 3 , 0.9 , 0.27 , ... }

a 1 = 10 , a n = 0.3 a n 1

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a n = { 0.61 , 1.83 , 5.49 , 16.47 , ... }

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a n = { 3 5 , 1 10 , 1 60 , 1 360 , ... }

a 1 = 3 5 , a n = 1 6 a n 1

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a n = { 2 , 4 3 , 8 9 , 16 27 , ... }

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a n = { 1 512 , 1 128 , 1 32 , 1 8 , ... }

a 1 = 1 512 , a n = 4 a n 1

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For the following exercises, write the first five terms of the geometric sequence.

a n = 4 5 n 1

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a n = 12 ( 1 2 ) n 1

12 , 6 , 3 , 3 2 , 3 4

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For the following exercises, write an explicit formula for each geometric sequence.

a n = { 2 , 4 , 8 , 16 , ... }

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a n = { 1 , 3 , 9 , 27 , ... }

a n = 3 n 1

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a n = { 4 , 12 , 36 , 108 , ... }

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a n = { 0.8 , 4 , 20 , 100 , ... }

a n = 0.8 ( 5 ) n 1

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a n = { 1.25 , 5 , 20 , 80 , ... }

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a n = { 1 , 4 5 , 16 25 , 64 125 , ... }

a n = ( 4 5 ) n 1

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a n = { 2 , 1 3 , 1 18 , 1 108 , ... }

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a n = { 3 , 1 , 1 3 , 1 9 , ... }

a n = 3 ( 1 3 ) n 1

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For the following exercises, find the specified term for the geometric sequence given.

Let a 1 = 4 , a n = 3 a n 1 . Find a 8 .

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Let a n = ( 1 3 ) n 1 . Find a 12 .

a 12 = 1 177 , 147

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For the following exercises, find the number of terms in the given finite geometric sequence.

a n = { 1 , 3 , 9 , ... , 2187 }

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a n = { 2 , 1 , 1 2 , ... , 1 1024 }

There are 12 terms in the sequence.

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Graphical

For the following exercises, determine whether the graph shown represents a geometric sequence.

Graph of a scattered plot with labeled points: (1, -0.5), (2, 0.25), (3, 1.375), (4, 3.0625), and (5, 5.5938). The x-axis is labeled n and the y-axis is labeled a_n.

The graph does not represent a geometric sequence.

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For the following exercises, use the information provided to graph the first five terms of the geometric sequence.

a 1 = 3 , a n = 2 a n 1

Graph of a scattered plot with labeled points: (1, 3), (2, 6), (3, 12), (4, 24), and (5, 48). The x-axis is labeled n and the y-axis is labeled a_n.
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a n = 27 0.3 n 1

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Extensions

Use recursive formulas to give two examples of geometric sequences whose 3 rd terms are 200.

Answers will vary. Examples: a 1 = 800 , a n = 0.5 a n 1 and a 1 = 12.5 , a n = 4 a n 1

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Use explicit formulas to give two examples of geometric sequences whose 7 th terms are 1024.

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Find the 5 th term of the geometric sequence { b , 4 b , 16 b , ... } .

a 5 = 256 b

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Find the 7 th term of the geometric sequence { 64 a ( b ) , 32 a ( 3 b ) , 16 a ( 9 b ) , ... } .

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At which term does the sequence { 10 , 12 , 14.4 , 17.28 ,   ... } exceed 100 ?

The sequence exceeds 100 at the 14 th term, a 14 107.

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At which term does the sequence { 1 2187 , 1 729 , 1 243 , 1 81   ... } begin to have integer values?

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For which term does the geometric sequence a n = 36 ( 2 3 ) n 1 first have a non-integer value?

a 4 = 32 3 is the first non-integer value

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Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10 th term.

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Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first 4 terms, and then find the 8 th term.

Answers will vary. Example: Explicit formula with a decimal common ratio: a n = 400 0.5 n 1 ; First 4 terms: 400 , 200 , 100 , 50 ; a 8 = 3.125

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Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.

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Questions & Answers

find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
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Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
Practice Key Terms 2

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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