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In this section, you will:
  • 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 of a 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 1 r , a 1 r 2 , a 1 r 3 , ... } .

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

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

Finding common ratios

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

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

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

  1. 2 1 = 2 4 2 = 2 8 4 = 2 16 8 = 2

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

  2. 12 48 = 1 4 4 12 = 1 3 2 4 = 1 2

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

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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.

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

5 , 10 , 15 , 20 , ...

The sequence is not geometric because 10 5 15 10 .

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

100 , 20 , 4 , 4 5 , ...

The sequence is geometric. The common ratio is 1 5 .

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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.

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.
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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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