Geometric Progression – GP Formulas

A geometric progression (GP) is one in which each term has a constant ratio to the one before it. It’s a unique kind of evolution. We must multiply with a set term known as the common ratio every time we want to find the next term in the geometric progression. Every time we need to get the previous term in the progression, we must divide the word to the same common ratio. Example: A GP with a common ratio of 2 is 2, 4, 8, 16, 32, etc. Finite or infinite geometric progressions are possible. Its common ratio might be either positive or negative. 

Finite or infinite geometric progressions are possible. Its common ratio might be either positive or negative.

Geometric Progression

“A geometric progression is one in which each term has a set ratio known as the common ratio”.GP is another name for it”.

The GP is usually written as a, ar, ar²,…, where a_n is the first term and r is the progression’s common ratio. Both negative and positive numbers are possible for the common ratio. We simply need the initial term and the constant ratio to find the terms of a geometric series.

Formulas for GP

To find the nth term in a progression, use the geometric progression formula. We need the first term and the common ratio to find the nth term. If you don’t know what the common ratio is, you can figure it out by multiplying any term by its previous term. The formula for the geometric progression’s nth term is:

a_n=ar^n-1

Where,

The first term is a

The common ratio is r

n is the number of the phrase we’re looking for.

Sum of Geometric Progression

To find the sum of all the terms in a geometric progression, use the geometric progression sum formula. Because geometric progression is divided into two types, finite and infinite geometric progressions, the sum of their terms is calculated using distinct formulas.

Finite Geometric Series

If a geometric progression has a finite number of terms, the total of the geometric series is determined using the formula:

S_n=a(1-rn) / (1-r)  For r≠1

Where,

The first term is a

The common ratio is r

 Number of terms in the series is n

Infinite Geometric Series

The infinite geometric series sum formula is employed when the number of terms in a geometric progression is unlimited. Depending on the value of r, two instances arise in infinite series.

if |r| < 1 then, 

S∞  =a / (1-r)

if |r| > 1 then,

The series does not converge and does not have a sum in this situation.

Conclusion

In this article we conclude that, a non-zero numeric series in which each term following the first is found by multiplying the preceding one by a fixed, non-zero amount known as the common ratio. We learn about this because we come across geometric sequences in real life and need a formula to assist us discover a certain number in the series. Our geometric sequence is defined as a set of integers, each of which is the preceding number multiplied by a constant.