<< Chapter < Page Chapter >> Page >

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

a 6 = 16 , 384

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write an explicit formula for the following geometric sequence.

{ –1 3 –9 27 ... }

a n = ( 3 ) n 1

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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] .
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask