Types of Geometric Progression

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. Every time we want to find the next term in the geometric sequence, we must multiply with a fixed term known as the common ratio 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 geometric series, each number, or term, is found by multiplying the previous term by a common ratio r. The geometric series can be expressed as follows if we call the first term a. A geometric progression (GP) is said to be finite if it contains finite terms. If a GP has infinite terms, it is called an infinite GP. In both series, the first term and the common ratio have the same meaning.

Finite geometric progression (Finite GP)

It’s a series with a finite number of terms, which means the series’ final term is defined. The general expression can be used to describe it.

For example, a finite geometric series with the last term is 1/2,1/4,1/8,1/16,…,1/32768.

Infinite geometric progression (Infinite GP)

The opposite of a finite geometric progression series is infinite geometric progression series. It’s an infinite series, so the series’ final term isn’t known.The general expression can be used to describe it.

For example, 3, -6, +12, -24, +… is an infinite series with no defined last term.

Representations and the formulas

Sum of Finite Geometric Progression

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

In geometric progression (also known as geometric series), the sum is given by

S = a₁ + a₂ + a₃ + a₄ ….. + a_n

S = a₁ + a₁r + a₁r² + a₁r³ +…+ a₁r^n-1……..(1)

When you multiply both sides of Equation (1) by r, we get

Sr = a₁r + a₁r² + a₁r³ + a₁r⁴ +… + a₁rⁿ …..(2)

Equation (2) is subtracted from Equation (1)

S – Sr = a₁ – a₁rⁿ

(1 – r)S = a₁(1-rⁿ)

S = a₁(1-rⁿ)/(1 – r)

The formula above is correct for GP with r < 1.0

When you subtract Equation (1) from Equation (2), we get

S = a₁(rⁿ-1)/(r-1)

Sum of Infinite Geometric Progression

An infinite geometric series sum formula is used when the number of terms in a geometric progression is infinite.

In an infinite geometric progression, the number of terms approaches infinity  (n = ∞). Only the range  -1.0 < (r ≠ 0) < +1.0 can be used to define the sum of infinite geometric progression.

S = a₁(1-rⁿ)/(1 – r)

S = (a₁-a₁rⁿ)/(1-r)

S = [a₁/(1-r)] – [a₁rⁿ/(1-r)]

For n→∞, the quantity [a₁rⁿ/(1-r)]→0 for -1.0<r≠0<+1.0

S = a₁/(1-r)

Conclusion

In this article we conclude that, 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. Every time we multiply with a fixed term known as the common ratio, we get the next term in the geometric progression. There are two types of geometric progression.  A geometric series is an infinite series whose terms follow a geometric progression or have a common ratio between them. The sum of the terms of a geometric series will be finite if the terms approach zero.