Geometric Progression (G.P.)

A geometric progression is also known as a geometric sequence in mathematics.  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 progression. We must multiply with a fixed term known as the common ratio every time we want to find the next term in the geometric progression, and we must divide the term with the same common ratio every time we want to find the preceding term in the progression.

In a sequence of numbers, if the ratio of any term to its preceding term is constant, the sequence is known as Geometric Progression.

It’s a collection of numbers obtained by multiplying or dividing the previous number by a constant number. The constant number is the series’ common ratio. A geometric progression is a collection of numbers with a common ratio. A positive or negative integer or fraction is the common ratio. It can be either a finite or infinite series.

Definition of Geometric Sequences

“A geometric sequence is a type of sequence in which the ratio of every two consecutive terms is fixed. This ratio is known as the geometric sequence’s common ratio. In other words, each term in a geometric sequence is multiplied by a constant, resulting in the next term”.

Geometric sequences are classified according to the number of terms they contain. 

• Finite geometric sequences: A geometric sequence with a finite number of terms is called a finite geometric sequence. That is, its final term is defined.

For example, a finite geometric sequence with the last term of 13122 is 2, 6, 18, 54,….13122.

• Infinite geometric sequences: A geometric sequence with an infinite number of terms is known as an infinite geometric sequence. That is, its final term is undefined. 

For example, 2, 4, 8, 16,… is an infinite sequence with no defined last term.

Formulas for geometric sequence

All geometric sequence formulas are listed below.

n^th term:  a_n = ar^n-1 or a_n = ra_n-1

First n terms’ sum: S_n = a(rⁿ-1)/(r-1)  when r ≠ 1 and S_n = na

Infinite terms sum: S_∞ = a/(1-r) , when |r| < 1

Example of geometric sequence

Calculate the sum of the first 18 terms in the geometric sequence 2, 6, 18, 54,…..

Solution:

a = 2 is the first term here.

r= 6/2 = 18/6 = 54/18 = 3 is the common ratio

n = 18  is number of terms

The formula for the sum of finite geometric sequences is:

S_n = a(rⁿ-1)/(r-1) 

S₁₈ = 2(3¹⁸-1)(3-1)

3¹⁸-1

The sum of the given geometric sequence’s first 18 terms is 3¹⁸-1

Properties of Geometric Progression

There are some properties of geometric progression that aid in the simple solution of mathematical problems. The following are a few of them:

• When each term of a geometric series is multiplied by a non-zero constant number, the result is a new series with the same common ratio.

• When we divide each term of the geometric series with the non-zero constant number, the new series also forms geometric series with the same common ratio.

• The reciprocal series of geometric progression terms is also a geometric series.

• The new series also forms geometric series when each term of the series changes to the terms’ square.

• The new series is geometric when the terms of a geometric progression are selected at intervals.

• Each term of geometric progression (non-zero, non-negative series) becomes an arithmetic series when the logarithm is changed.

Conclusion

We looked at the definition and properties of geometric progression in this article, which is a sequence of numbers connected by a common ratio. The first term, common ratio, and nth term of the geometric progression, as well as the formulas, have all been covered. We learned about finite and infinite series, as well as the formulas for calculating the geometric progression’s sum of terms. The geometrical progression and some types of problems were discussed in this article. Using solved examples, some of the concepts are explained.