# Geometric progressions

A sequence of numbers when the ratio of any two consecutive terms is the same is known as geometric progressions. To understand this easily, we can say that in geometric progressions the next number in a series can be calculated by multiplying a fixed number by the previous number.

## Introduction

In a geometrical progression, the next number in a series can be found by simply multiplying the previous digit with a constant number. For example, 2, 4, 8, 16 can be called a geometric progression because the ratio of two continuing terms in the series remains constant.

That is, 4/2 = 8/2 = 16/2 = 2.

## Common facts about Geometric Progression

In a geometric progression, a certain common fact can be found. These include:

• The first number in a geometric progression is known as the initial term
• A common ratio is a term used to define the ratio between a number in the sequence and the number before it
• We can determine the common ratio in a geometric progression by simply dividing any number by its preceding number
• If the common ratio is positive then all the terms will have the same sign as the starting term
• If the common ratio is negative then the terms will keep changing from positive and negative
• Reciprocal of all the terms in the geometric progression also form a geometric progression

These are certain points that can help in enhancing the understanding of geometric progressions

## General Example to explain the geometric sequence

In a sequence, powers rk of a fixed non-zero number, for example, 2k and 3k. This would form the basic geometric sequence as:

A, Ar2, Ar3, Ar4….and so on

Where,

• r is not equal to zero would be the common ratio
• A is not equal to zero is a scale factor

## The formula of the nth term of geometric progression

If ‘Z’ is the first term and ‘r’ is the common ratio then the formula would be:

Zn = Z1 * rn-1

Where,

Zn = General term

Z1 = First Term

rn-1 = Common Ratio

This formula can help in determining the nth term of geometric progression.

## The formula of the sum of the nth term of geometric progression

Sum = a (rn – 1) / r – 1

Where,

r = Common ratio

n = Number of terms

Sum = Sum of all geometric progression

This formula can help in determining the sum of the nth term of geometric progression.

### Conclusion

Geometric progression refers to a sequence of numbers in which the next term in the series can be calculated by multiplying a fixed number to the preceding number in the series. For example, the sequence 1, 2, 4, 8… can be said that as a geometric progression with a common ratio of 2. In a geometric progression, the common ratio plays a pivotal role in determining whether the sequence is in geometric progression or not. The common ratio can be easily calculated by dividing a term with its preceding term in the sequence and if throughout the sequence this ratio remains constant then it is known as the common ratio.