### Geometric series

A geometric series is a series whose related sequence is geometric. It results from adding the terms of a geometric sequence

In General we write a Geometric Sequence like this:

{a, ar, ar^{2}, ar^{3}, … }

Where,

- a is the first term, and
- r is the factor between the terms (called the “common ratio”)

### Nth Term of GP:

a_{n} = ar^{(n-1)}

### Example:

If 2,4,8,…., is the GP, then find its 10th term.

### Solution:

The nth term of GP is given by

a_{n} = ar^{(n-1)}

Here, a = 2 and r = 4/2 = 2

Therefore,

a_{10} = 2 x 2^{2-1}

a_{10} = 2 x 2

a_{10} = 4

### Sum of N term of GP:

$$ S_n= a\left(\frac {1-r^n}{1-r}\right) $$

### Example:

Sum the first 4 terms of 10, 30, 90, 270, 810, 2430, …

### Solution:

The values of a, r and n are:

- a = 10 (the first term)
- r = 3 (the “common ratio”)
- n = 4 (we want to sum the first 4 terms)

$$ S_4= 10\left(\frac {1-3^4}{1-3}\right) $$

$$ S_4= 400 $$