<< Chapter < Page Chapter >> Page >

Section exercises

Verbal

What is a geometric sequence?

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

Got questions? Get instant answers now!

How is the common ratio of a geometric sequence found?

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Algebraic

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

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

Got questions? Get instant answers now!

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

The common ratio is 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

The sequence is geometric. The common ratio is 5.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

a 4 = 16 27

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

a 7 = 2 729

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

7 , 1.4 , 0.28 , 0.056 , 0.0112

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

a n = 4 5 n 1

Got questions? Get instant answers now!

a n = 12 ( 1 2 ) n 1

12 , 6 , 3 , 3 2 , 3 4

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

a n = 3 n 1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

a n = 0.8 ( 5 ) n 1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

a n = ( 4 5 ) n 1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

a n = 3 ( 1 3 ) n 1

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

a 12 = 1 177 , 147

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

There are 12 terms in the sequence.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

a n = 27 0.3 n 1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

a 5 = 256 b

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask