Geometric progression

Geometric Progression, Series & Sums

Introduction

A geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. The common ratio (r) is obtained by dividing any term by the preceding term, i.e.,

where	r	common ratio
	a1	first term
	a2	second term
	a3	third term
	an-1	the term before the n th term
	an	the n th term

The geometric sequence is sometimes called the geometric progression or GP, for short.

For example, the sequence 1, 3, 9, 27, 81 is a geometric sequence. Note that after the first term, the next term is obtained by multiplying the preceding element by 3.

The geometric sequence has its sequence formation: 

To find the nth term of a geometric sequence we use the formula:

where	r	common ratio
	a1	first term
	an-1	the term before the n th term
	n	number of terms

Sum of Terms in a Geometric Progression

Finding the sum of terms in a geometric progression is easily obtained by applying the formulas:

nth partial sum of a geometric sequence

sum to infinity

where	Sn	sum of GP with n terms
	S∞	sum of GP with infinitely many terms
	a1	the first term
	r	common ratio
	n	number of terms