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.
Write a formula for the student population.
Estimate the student population in 2020.
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
$${P}_{n}=284\cdot {1.04}^{n}$$
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.
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.
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
$\text{\hspace{0.17em}}{a}_{n}=r{a}_{n\u20131}\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}{a}_{n}={a}_{1}{r}^{n\u20131}.$ 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] .
Questions & Answers
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.