In real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[/latex] instead of [latex]{a}_{1}[/latex]. In these problems, we can alter the explicit formula slightly by using the following formula:
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 [latex]P[/latex] be the student population and [latex]n[/latex] be the number of years after 2013. Using the explicit formula for a geometric sequence we get
[latex]{P}_{n} =284\cdot {1.04}^{n}[/latex] - We can find the number of years since 2013 by subtracting.
[latex]2020 - 2013=7[/latex]
We are looking for the population after 7 years. We can substitute 7 for [latex]n[/latex] to estimate the population in 2020.
[latex]{P}_{7}=284\cdot {1.04}^{7}\approx 374[/latex]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.