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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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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