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Given a geometric sequence with a 2 = 4 and a 3 = 32 , find a 6 .

a 6 = 16 , 384

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Writing an explicit formula for the n Th term of a geometric sequence

Write an explicit formula for the n th 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.

10 2 = 5

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

a n = a 1 r ( n 1 ) a n = 2 5 n 1

The graph of this sequence in [link] shows an exponential pattern.

Graph of the geometric sequence.
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Write an explicit formula for the following geometric sequence.

{ –1 3 –9 27 ... }

a n = ( 3 ) n 1

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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 0 r n

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.

  1. Write a formula for the student population.
  2. Estimate the student population in 2020.
  1. 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 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
  2. We can find the number of years since 2013 by subtracting.

    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.

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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.

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

  1. P n   =   293 1.026 a n
  2. The number of hits will be about 333.

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

Key equations

recursive formula for n t h term of a geometric sequence a n = r a n 1 , n 2
explicit formula for n t h term of a geometric sequence a n = a 1 r n 1

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 [link] .
  • The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See [link] and [link] .
  • A recursive formula for a geometric sequence with common ratio r is given by a n = r a n 1 for n 2 .
  • As with any recursive formula, the initial term of the sequence must be given. See [link] .
  • An explicit formula for a geometric sequence with common ratio r is given by a n = a 1 r n 1 . See [link] .
  • In application problems, we sometimes alter the explicit formula slightly to a n = a 0 r n . See [link] .

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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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