# Geometric Progression (GP)

## What is a geometric progression

A geometric progression (GP) is one in which the ratio (r) between any 2 consecutive terms is a constant.

## Examples of geometric progression

An geometric progression is one where the ratio between any 2 consecutive terms is a constant. This ratio can be a positive or negative difference. Examples of arithmetic progressions are:

• 2, 4, 8, 16, 32, 64….. In this example here, the first term is 2, and to get to the next term, we multiply it by 2. We call 2 the common ratio of this geometric progression.
• 20, -10, 5, -2.5, …… In this example here, the first term is 20, and to get to the next term, we multiply it by -0.5. We call -0.5 the common ratio of this geometric progression.

## General Expression for nth term and sum of first n terms

If a is the first term, and r is the common ratio of an geometric progression,

nth term, uₙ = arⁿ⁻¹

Sum of first n terms, Sₙ is given by:

## How to prove a sequence is a geometric progression?

Since a geometric progression is one where the ratio between 2 consecutive terms is a constant, to prove an geometric progression, you’ll need to prove that uₙ/uₙ₋₁ = constant.

## Sum to infinity of a geometric progression

When |r| < 1, as n →∞, uₙ → 0.

Hence, a geometric progression is convergent only when |r| < 1. When that happens, a sum to infinity exists, and the sum to infinity, S∞ is given by:

## All the notes for H2 A Level Math

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