# Geometric Sequences

How to find the nth 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).

a2/a1 = r
a3/a2 = r
a4/a3 = r
...

r is the [common ratio].

So to find the next term,
multiply the common ratio: × r.

The nth term of a geometric sequence, an,
can be found by using the formula below.

an = a1rn - 1

an: nth term
a1: First term
r: Common ratio

## Example 1

The first term is 2.
So a1 = 2.

6/2 = 3
18/6 = 3
54/18 = 3
162/54 = 3
...
So r = 3.

So an = 2⋅3n - 1

## Example 2

The first term is 320.
So a1 = 320.

320/160 = 1/2
160/80 = 1/2
80/40 = 1/2
...
So r = 1/2.

So ak = 320⋅(1/2)k - 1

320 = 26⋅5

Prime factorization

(1/2)k - 1 = 2-k + 1

Negative exponent

26⋅2-k + 1 = 2-k + 7

Product of powers

ak = 5⋅2-k + 7

And it says
ak = 5/8.

So 5⋅2-k + 7 = 5/8.

5/8
= 5/23
= 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

a5 = a1r = -6
a5 = a1r4 = 48

So
a1r = -6,
a1r4 = 48.

The goal is to find a1 and r.

a1r4 = 48
a1r = -6

To eliminate a1,
set a1r4/a1r.

Then a1r4/a1r = 48/(-6).

Cancel a1r
in both of the numerator and the denominator.

Then r3 = -8.

-8 = (-2)3

So r = -2.

The exponent of r3 is 3:
an odd exponent.
So you don't need to write [±]
in front of (-2).

Put [r = -2] into [a1r = -6].

Then a1⋅(-2) = -6.

Divide both sides by -2.

Then a1 = 3.

a1 = 3
r = -2

So an = 3⋅(-2)n - 1.