# Geometric Sequence

How to find the nth term of a geometric sequence: definition, formula, 3 examples, and their solutions.

## Definition

### Definition

For the terms of a sequence,
if you multiply a term and a constant r
then you get the next term.

a1⋅r = a2
a2⋅r = a3
a3⋅r = a4
...

This sequence is a geometric sequence.

This r is called the common ratio.
It is the ratio of the adjacent terms.
(a2/a1 = r, a3/a2 = r, ...)

## Formula

### Formula

For a geometric sequence,
an = arn - 1.

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

## Example 1

### Solution

Find the first term a and r.

Write 2, 6, 18, 54, 162.

The first term, a, is 2.

2⋅3 = 6
6⋅3 = 18
18⋅3 = 54
54⋅3 = 162

So write, the r, ×3
between the terms.

a = 2
r = 3

So an = 2⋅3n - 1.

So an = 2⋅3n - 1.

## Example 2

### Solution

Find the first term a and r.

Write 320, 160, 80, 40.

The first term, a, is 320.

320⋅[1/2] = 160
160⋅[1/2] = 80
80⋅[1/2] = 40

So write, the r, ×[1/2]
between the terms.

a = 320
r = 1/2

So ak = 320⋅[1/2]k - 1.

26⋅2-k + 1
= 26 + (-k + 1)
= 2-k + 7

Product of Powers

So ak = 2-k + 7⋅5.

It says
ak = 5/8.

And you found
ak = 2-k + 7⋅5.

So
2-k + 7⋅5 = 5/8.

5/8
= 5/23
= 5⋅2-3

Negative Exponent

Cancel the factors 5 on both sides.

The bases of both sides are the same: 2.

Then the exponents of both sides are the same.

So -k + 7 = -3.

Exponential Equation

Move +7 to the right side.

Then -k = -10.

Divide both sides by -1.

Then k = 10.

So k = 10.

## Example 3

### Solution

To find an,
find the first term a
and the common ratio r.

For a geometric sequence,
a2 = a⋅r1.

And it says
a2 = -6.

So a2 = ar1 = -6.

For a geometric sequence,
a5 = a⋅r4.

And it says
a5 = 48.

So a5 = ar4 = 48.

ar1 = -6
ar4 = 48

The goal is to solve this system
and find the values of a and r.

To cancel a,
set a proportion like this.

ar4/ar1 = 48/(-6)

ar4/ar1
= r4/r1
= r4 - 1
= r3

Negative Exponent

48/(-6) = -8

-8 = (-2)3

r3 = (-2)3

Cube root both sides.

Then r = -2.

r = -2

Put this into ar1 = -6.

Then a⋅(-2) = -6.

Substitution Method

Divide both sides by -2.

Then a = 3.

a = 3
r = -2

So an = 3⋅(-2)n - 1.

So an = 3⋅(-2)n - 1.