Geometric Progression Concepts

Understanding the Concepts of Geometric Progression

Geometric Progression (GP) is a sort of sequence in which each succeeding term is produced by multiplying the previous term by a fixed number known as a common ratio.

This progression is also known as a pattern-following geometric sequence of numbers.

The common ratio multiplied by each term to produce the next term is not zero.

For Example

A geometric series consists of the numbers 2, 4, 8, 16, 32, 64,..., where the common ratio is 2.

In general, it is expressed as:

a, ar, ar2, ar3,......arn-1, and so on.

Where a represents the first phrase and r represents the common ratio.

The formula below can be used to get the Nth term of a G.P.

Similarly, nth term, an =arn-1

1. GP Common Ratio

Consider the sequence a, ar, ar2,ar3,......arn-1. The common ratio can be found by dividing every successive term by its preceding term.

Suppose we divide the 3rd term by the 2nd term we get:

ar2/ar = r

In a similar manner:

ar3/ar2 = r

ar4/ar3 = r

In general, the common ratio is defined= (Any term) / (Preceding term)

2. Sum of N term of GP

Suppose, ar, ar2, ar3,……

The provided Geometric Progression is arn-1.

The total of n terms of GP is thus given by:

Sn = a + ar + ar2 + ar3+…+ arn-1

The formula for calculating the sum of n terms of GP is as follows:

Sn = a[(rn – 1)/(r – 1)] if r ≠ 1 and r > 1

Where

a - First term

d - Common difference

n - Number of terms

3. Types of Geometric Progression

Based on the number of terms, geometric progression can be classified into two categories. They are as follows:

1. Finite geometric progression (Finite GP)
2. Infinite geometric progression (Infinite GP)

These two GPs are detailed below, along with their representations and formulas for calculating the sum.

1. Finite Geometric Progression

A finite G.P.'s terms are expressed as a, ar, ar2,ar3,......arn-1.

arn-1 is referred to as a finite geometric series.

The sum of finite Geometric series can be calculated as follows:

Sn = a[(rn – 1)/(r – 1)] if r ≠ 1 and r > 1

2. Infinite Geometric Progression

An infinite G.P. can be expressed as a, ar, ar2,ar3,......arn-1.

The total of infinite geometric series can be calculated as follows:

Sn = a[(rn – 1)/(r – 1)] if r ≠ 1 and r > 1

This is called the geometric progression formula of sum to infinity.