Geometric progressions

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.