# Geometric Sequences

How to find the *n*th term of a geometric sequence: formula, examples, and their solutions.

## Formula

A geometric sequence is a sequence

whose ratios of the adjacent terms

are the same (= *r*).*a*_{2}/*a*_{1} = *r**a*_{3}/*a*_{2} = *r**a*_{4}/*a*_{3} = *r*

...*r* is the [common ratio].

So to find the next term,

multiply the common ratio: × *r*.

The *n*th term of a geometric sequence, *a*_{n},

can be found by using the formula below.*a*_{n} = *a*_{1}⋅*r*^{n - 1}*a*_{n}: *n*th term*a*_{1}: First term*r*: Common ratio

## Example 1

The first term is 2.

So *a*_{1} = 2.

6/2 = 3

18/6 = 3

54/18 = 3

162/54 = 3

...

So *r* = 3.

So *a*_{n} = 2⋅3^{n - 1}

## Example 2

The first term is 320.

So *a*_{1} = 320.

320/160 = 1/2

160/80 = 1/2

80/40 = 1/2

...

So *r* = 1/2.

So *a*_{k} = 320⋅(1/2)^{k - 1}

320 = 2^{6}⋅5

Prime factorization

(1/2)^{k - 1} = 2^{-k + 1}

Negative exponent

2^{6}⋅2^{-k + 1} = 2^{-k + 7}

Product of powers

*a*_{k} = 5⋅2^{-k + 7}

And it says*a*_{k} = 5/8.

So 5⋅2^{-k + 7} = 5/8.

5/8

= 5/2^{3}

= 5⋅2^{-3}

Divide both sides by 5.

Then 2^{-k + 7} = 2^{-3}.

2^{-k + 7} = 2^{-3}

The bases are the same.

So -*k* + 7 = -3.

Exponential equations

Move +7 to the right side.

Then -*k* = -10.

Divide both sides by -1.

Then *k* = 10.

## Example 3

*a*_{5} = *a*_{1}⋅*r* = -6*a*_{5} = *a*_{1}⋅*r*^{4} = 48

So*a*_{1}⋅*r* = -6,*a*_{1}⋅*r*^{4} = 48.

The goal is to find *a*_{1} and *r*.

*a*_{1}⋅*r*^{4} = 48*a*_{1}⋅*r* = -6

To eliminate *a*_{1},

set *a*_{1}⋅*r*^{4}/*a*_{1}⋅*r*.

Then *a*_{1}⋅*r*^{4}/*a*_{1}⋅*r* = 48/(-6).

Cancel *a*_{1}⋅*r*

in both of the numerator and the denominator.

Then *r*^{3} = -8.

-8 = (-2)^{3}

So *r* = -2.

The exponent of *r*^{3} is 3:

an odd exponent.

So you don't need to write [±]

in front of (-2).

Put [*r* = -2] into [*a*_{1}⋅*r* = -6].

Then *a*_{1}⋅(-2) = -6.

Divide both sides by -2.

Then *a*_{1} = 3.

*a*_{1} = 3*r* = -2

So *a*_{n} = 3⋅(-2)^{n - 1}.