# Geometric sequences review

CCSS Math: HSF.LE.A.2
Review geometric sequences and solve various problems involving them.

## Parts and formulas of geometric sequence

In geometric sequences, the ratio between consecutive terms is always the same. We call that ratio the common ratio.
For example, the common ratio of the following sequence is $2$:
$\footnotesize\maroonC{\times 2\,\Large\curvearrowright}$$\footnotesize\maroonC{\times 2\,\Large\curvearrowright}$$\footnotesize\maroonC{\times 2\,\Large\curvearrowright}$
$1,$$2,$$4,$$8,...$
Geometric sequence formulas give $a(n)$, the $n^{\text{th}}$ term of the sequence.
This is the explicit formula for the geometric sequence whose first term is $\blueD k$ and common ratio is $\maroonC r$:
$a(n)=\blueD k\cdot\maroonC r^{n-1}$
This is the recursive formula of that sequence:
$\begin{cases}a(1) = \blueD k \\\\ a(n) = a(n-1)\cdot\maroonC r \end{cases}$

## Extending geometric sequences

Suppose we want to extend the sequence $54,18,6,...$ We can see each term is $\maroonC{\times\dfrac{1}{3}}$ from the previous term:
$\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}$$\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}$
$54,$$18,$$6,...$
So we simply multiply that ratio to find that the next term is $2$:
$\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}$$\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}$$\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}$
$54,$$18,$$6,$$2,...$
Problem 1
What is the next term in the sequence $\dfrac{1}{2},2,8, \ldots$?
Want to try more problems like this? Check out this exercise.

## Writing recursive formulas

Suppose we want to write a recursive formula for $54,18,6,...$ We already know the common ratio is $\maroonC{\times\dfrac{1}{3}}$. We can also see that the first term is $\blueD{54}$. Therefore, this is a recursive formula for the sequence:
$\begin{cases}a(1) = \blueD{54} \\\\ a(n) = a(n-1)\cdot\maroonC{\dfrac{1}{3}} \end{cases}$
Problem 1
Find $k$ and $r$ in this recursive formula of the sequence $\dfrac{1}{2},2,8, \ldots$.
$\begin{cases}a(1) = k \\\\ a(n) = a(n-1)\cdot r \end{cases}$
$k=$
$r=$
Want to try more problems like this? Check out this exercise.

## Writing explicit formulas

Suppose we want to write an explicit formula for $54,18,6,...$ We already know the common ratio is $\maroonC{\times\dfrac{1}{3}}$ and the first term is $\blueD{54}$. Therefore, this is an explicit formula for the sequence:
$a(n)=\blueD{54}\cdot\left(\maroonC{\dfrac{1}{3}}\right)^{n-1}$
Problem 1
Write an explicit formula for $\dfrac{1}{2},2,8, \ldots$
$a(n)=$
Want to try more problems like this? Check out this exercise.