# 13.3 Geometric sequences  (Page 3/6)

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Given a geometric sequence with ${a}_{2}=4$ and ${a}_{3}=32$ , find ${a}_{6}.$

${a}_{6}=16,384$

## Writing an explicit formula for the n Th term of a geometric sequence

Write an explicit formula for the $n\text{th}$ term of the following geometric sequence.

The first term is 2. The common ratio can be found by dividing the second term by the first term.

$\frac{10}{2}=5$

The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.

$\begin{array}{l}{a}_{n}={a}_{1}{r}^{\left(n-1\right)}\hfill \\ {a}_{n}=2\cdot {5}^{n-1}\hfill \end{array}$

The graph of this sequence in [link] shows an exponential pattern.

Write an explicit formula for the following geometric sequence.

${a}_{n}=-{\left(-3\right)}^{n-1}$

## 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 ${a}_{1}.\text{\hspace{0.17em}}$ In these problems, we can alter the explicit formula slightly by using the following formula:

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

## 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.
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 $P$ be the student population and $n$ be the number of years after 2013. Using the explicit formula for a geometric sequence we get

2. 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.

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.

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

1. The number of hits will be about 333.

Access these online resources for additional instruction and practice with geometric sequences.

## Key equations

 recursive formula for $nth$ term of a geometric sequence ${a}_{n}=r{a}_{n-1},n\ge 2$ explicit formula for $\text{\hspace{0.17em}}nth\text{\hspace{0.17em}}$ term of a geometric sequence ${a}_{n}={a}_{1}{r}^{n-1}$

## Key concepts

• A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
• The constant ratio between two consecutive terms is called the common ratio.
• The common ratio can be found by dividing any term in the sequence by the previous term. See [link] .
• The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See [link] and [link] .
• A recursive formula for a geometric sequence with common ratio $r$ is given by $\text{\hspace{0.17em}}{a}_{n}=r{a}_{n–1}\text{\hspace{0.17em}}$ for $n\ge 2$ .
• As with any recursive formula, the initial term of the sequence must be given. See [link] .
• An explicit formula for a geometric sequence with common ratio $r$ is given by $\text{\hspace{0.17em}}{a}_{n}={a}_{1}{r}^{n–1}.$ See [link] .
• In application problems, we sometimes alter the explicit formula slightly to $\text{\hspace{0.17em}}{a}_{n}={a}_{0}{r}^{n}.\text{\hspace{0.17em}}$ See [link] .

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