Geometric series

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, ar2, ar3, … }

Where,

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

an = 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

an = ar(n-1)

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

Therefore,

a10 = 2 x 22-1

a10 = 2 x 2

a10 = 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)

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

$$S_4= 400$$