# Geometric Series

How to solve the geometric sequence problems: formula, proof, examples, and their solutions.

## Formula

A geohmetric series is the sum

of the geometric sequences' terms.*S*_{n} = *a*_{1}(1 - *r*^{n}) / (1 - *r*)*S*_{n}: the sum of '1st term ~ *n*th term'*a*_{1}: first term*r*: common ratio

## Proof

*S*_{n} = *a*_{1} + *a*_{2} + *a*_{3} + ... + *a*_{n}

= *a*_{1} + *a*_{1}*r* + *a*_{1}*r*^{2} + ... + *a*_{1}*r*^{n - 1}

Geometric sequences

Write *rS*_{n} on the next line.*rS*_{n} = *a*_{1}*r* + *a*_{1}*r*^{2} + *a*_{1}*r*^{3} + ... + *a*_{1}*r*^{n}

Subtract these two.

Then (1 - *r*)*S*_{n} = *a*_{1} - *a*_{1}*r*^{n}.

Other terms on the right side are all cancelled.

Divide both sides by (1 - *r*).

Then *S*_{n} = *a*_{1}(1 - *r*^{n}) / (1 - *r*).

## Example 1

*a*_{1} = 3, *r* = 2, *d* = 7*S*_{7} = 3(1 - 2^{7}) / (1 - 2)

## Example 2

*a*_{3} = 4⋅*r*^{2} = 36

So *r* = ±3.

Geometric sequences

Case 1) *r* = 3*S*_{5} = 4(1 - 3^{5}) / (1 - 3)

= 484

Case 2) *r* = -3*S*_{5} = 4(1 - (-3)^{5}) / [1 - (-3)]

= 244

So *S*_{5} = 484 or 244.

## Example 3

*a*_{n} = 5⋅(2/3)^{n}

So the sigma notation shows the geometric series.

To use the geometric series formula,

find *a*_{1}, *r*, and *n*.

Geometric sequences

*a*_{1} = 10/3, *r* = 2/3, *n* = 4

(given) = *S*_{4}

= (10/3)⋅[1 - (2/3)^{4}] / (1 - 2/3)

Multiply 3 to both of the numerator and the denominator.