# Geometric Sequence

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

## Definition

## Formula

For a geometric sequence,

a_{n} = ar^{n - 1}.

a_{n}: nth term

a: First term, a_{1}

r: Common ratio

## Example2, 6, 18, 54, 162, ..., a_{n} = ?

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 a_{n} = 2⋅3^{n - 1}.

So a_{n} = 2⋅3^{n - 1}.

## Example320, 160, 80, 40, ..., a_{k} = 5/8, k = ?

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 a_{k} = 320⋅[1/2]^{k - 1}.

320 = 2^{6}⋅5

Prime Factorization

[1/2]^{k - 1}

= 2^{-(k - 1)}

= 2^{-k + 1}

Negative Exponent

Multiply a Monomial and a Polynomial

2^{6}⋅2^{-k + 1}

= 2^{6 + (-k + 1)}

= 2^{-k + 7}

Product of Powers

So a_{k} = 2^{-k + 7}⋅5.

It says

a_{k} = 5/8.

And you found

a_{k} = 2^{-k + 7}⋅5.

So

2^{-k + 7}⋅5 = 5/8.

5/8

= 5/2^{3}

= 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.

## Examplea_{2} = -6, a_{5} = 48, a_{n} = ?

To find a_{n},

find the first term a

and the common ratio r.

For a geometric sequence,

a_{2} = a⋅r^{1}.

And it says

a_{2} = -6.

So a_{2} = ar^{1} = -6.

For a geometric sequence,

a_{5} = a⋅r^{4}.

And it says

a_{5} = 48.

So a_{5} = ar^{4} = 48.

ar^{1} = -6

ar^{4} = 48

The goal is to solve this system

and find the values of a and r.

To cancel a,

set a proportion like this.

ar^{4}/ar^{1} = 48/(-6)

ar^{4}/ar^{1}

= r^{4}/r^{1}

= r^{4 - 1}

= r^{3}

Negative Exponent

48/(-6) = -8

-8 = (-2)^{3}

r^{3} = (-2)^{3}

Cube root both sides.

Then r = -2.

r = -2

Put this into ar^{1} = -6.

Then a⋅(-2) = -6.

Substitution Method

Divide both sides by -2.

Then a = 3.

a = 3

r = -2

So a_{n} = 3⋅(-2)^{n - 1}.

So a_{n} = 3⋅(-2)^{n - 1}.