# Geometric sequences review

CCSS Math: HSF.BF.A.2, HSF.IF.A.2, HSF.IF.A.3, 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 $\maroonC r$:This is the

**recursive**formula of that sequence:*Want to learn more about geometric sequences? Check out this video.*

## 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 $2$:

$\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}$ | $\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}$ | $\maroonC{\times\dfrac{1}{3}\,\Large\curvearrowright}$ | ||||
---|---|---|---|---|---|---|

$54,$ | $18,$ | $6,$ | $2,...$ |

*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:

*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:

*Want to try more problems like this? Check out this exercise.*