Determine if a sequence is geometric, find the common ratio, list the terms, and find the general (nth) term of a geometric sequence
Use recursive and explicit formulas to describe and study geometric sequences
Solving Application Problems with Geometric Sequences
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:
[latex]{a}_{n}={a}_{0}{r}^{n}[/latex]
In 2021, the number of students in a small school is [latex]284[/latex]. It is estimated that the student population will increase by [latex]4 \%[/latex] each year.
Write a formula for the student population.
Estimate the student population in 2028.
The situation can be modeled by a geometric sequence with an initial term of [latex]284[/latex]. The student population will be [latex]104 \%[/latex] of the prior year, so the common ratio is [latex]1.04[/latex].Let [latex]P[/latex] be the student population and [latex]n[/latex] be the number of years after 2021. 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 2021 by subtracting.
[latex]2028 - 2021=7[/latex]
We are looking for the population after [latex]7[/latex] years. We can substitute [latex]7[/latex] for [latex]n[/latex] to estimate the population in 2028.
The student population will be about [latex]374[/latex] in 2028.
A new investor places [latex]$5,000[/latex] in a high-yield savings account that offers [latex]6 \%[/latex] annual interest, compounded annually. Assuming she doesn’t make any additional deposits or withdrawals, how much money will be in the account after [latex]10[/latex] years?
Identify the geometric sequence:
Initial amount [latex]a_1 = $5,000[/latex]
Common ratio [latex]r = 1 + 0.06 = 1.06[/latex] ([latex]106 \%[/latex] of previous year’s amount)
We want to find the [latex]10[/latex]th term [latex](n = 10)[/latex]
Therefore, after [latex]10[/latex] years, the account will contain approximately [latex]$8,447.50[/latex].
A new car is purchased for [latex]$32,000[/latex]. Each year, its value decreases by [latex]15 \%[/latex] of its value from the previous year. What will be the car’s value after [latex]5[/latex] years?
Identify the geometric sequence:
Initial value [latex]a_1 = $32,000[/latex]
Common ratio [latex]r = 1 - 0.15 = 0.85[/latex] ([latex]85 \%[/latex] of previous year’s value)
We want to find the [latex]5[/latex]th term [latex](n = 5)[/latex]