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Study Guides > College Algebra

Solving Application Problems with Geometric Sequences

In real-world scenarios involving arithmetic sequences, we may need to use an initial term of a0{a}_{0} instead of a1{a}_{1}. In these problems, we can alter the explicit formula slightly by using the following formula:

an=a0rn{a}_{n}={a}_{0}{r}^{n}

Example 7: 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.

Solution

  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 PP be the student population and nn be the number of years after 2013. Using the explicit formula for a geometric sequence we get
    Pn=2841.04n{P}_{n} =284\cdot {1.04}^{n}
  2. We can find the number of years since 2013 by subtracting.
    20202013=72020 - 2013=7
    We are looking for the population after 7 years. We can substitute 7 for nn to estimate the population in 2020.
    P7=2841.047374{P}_{7}=284\cdot {1.04}^{7}\approx 374
    The student population will be about 374 in 2020.

Try It 7

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.
a. Write a formula for the number of hits.
b. Estimate the number of hits in 5 weeks.
Solution

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  • Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution.