# 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] .

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
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