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}

${a}_{n}={a}_{0}{r}^{n}$

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.

Write a formula for the student population.
Estimate the student population in 2020.

Solution

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

${P}_{n} =284\cdot {1.04}^{n}$
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.

${P}_{7}=284\cdot {1.04}^{7}\approx 374$

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