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”)

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,

a₁₀ = 2 x 2^2-1

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)

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

$$ S_4= 400 $$