Use a Formula for a Geometric Sequence

Finally, we will find the terms of a geometric sequence given a recursive formula.

Solving Application Problems with Geometric Sequences

In real-world scenarios involving geometric 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:

$a_n=a_0r^n$

Example 6

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

$P_n =284⋅1.04^n$

ⓑ 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.

$P_7=284⋅1.04^7 \approx 374$

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.

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