In **geometric progression (G.P.)**, the sequence is geometric and is a result of the **sum of G.P**. A geometric series is the sum of all the terms of geometric sequence. Before going to learn how to find the sum of a given Geometric Progression, first know what a GP is in detail.

## What is Geometric Progression?

If in a sequence of terms, each succeeding term is generated by multiplying each preceding term with a constant value, then the sequence is called a **geometric progression**. (GP), whereas the constant value is called the common ratio. For example, 2, 4, 8, 16, 32, 64, … is a GP, where the common ratio is 2.

Similarly,

In the given examples, the ratio is a constant. Such sequences are called Geometric Progressions. It is abbreviated as G.P.

Where r is a constant which is known as common ratio and none of the terms in the sequence is zero.

Now, learn how to add GP if there are n number of terms present in it.

**Also check:** Arithmetic Progression

## Sum of Nth terms of G.P.

A geometric series is a sum of an infinite number of terms such that the ratio between successive terms is constant. In this section, we will learn to find the sum of geometric series.

### Derivation of Sum of GP

Since, we know, in a G.P., the common ratio between the successive terms is constant, so we will consider a geometric series of finite terms to derive the formula to find the sum of Geometric Progression.

Consider the G.P, a, ar, ar^{2}, ….ar^{n-1}.

Let S_{n}, a, r be the sum of n terms, first term and the ratio of the G.P respectively.

There are two cases, either r = 1 or r ≠ 1.

If r=1, then S_{n} = a + a + a + ⋯ a= na

When r ≠ 1,

Multiply (1) with r gives,

Subtracting (1) from (2) gives

Sum of n terms
\(\begin{array}{l}S_{n} = \frac{a(r^n – 1)}{r-1}; \ Where \ r \neq 1\end{array} \) |

The above formula is also called Geometric Progression formula or G.P. formula to find the sum of GP of finite terms. Here, r is the common ratio of G.P. formula.

## Sum of GP for Infinite Terms

If the number of terms in a GP is not finite, then the GP is called infinite GP. The formula to find the sum to infinity of the given GP is:

Here,

S_{∞} = Sum of infinite geometric progression

a = First term of G.P.

r = Common ratio of G.P.

n = Number of terms

This formula helps in converting a recurring decimal to the equivalent fraction. This can be observed from the following example:

0.22222222… = 0.2 + 0.02 + 0.002 + 0.0002 + …..∞

= (0.2 × 0.1^{0}) + (0.2 × 0.1^{1}) + (0.2 × 0.1^{2}) + (0.2 × 0.1^{3}) + ….∞

= (0.2) + (0.2 × 0.1) + (0.2 × 0.1^{2}) + (0.2 × 0.1^{3}) + ….∞

This of the form a + ar + ar^{2} + ar^{3} + ….. ∞ (infinite GP) such that a = 0.2 and r = 0.1.

Thus, 0.22222222… = 0.2/(1 – 0.1)

= 0.2/0.9

= 2/9

Hence, the equivalent fraction of 0.22222222… is 2/9.

## Video Lesson

### Sum of Infinite Terms of G.P.

## Solved Examples on Sum of G.P.

**Example 1**: If n

^{th}term of the G.P 3, 6, 12, …. is 192, then what is the value of n?

**Solution**: First, we have to find the common ratio

n = 7

Therefore, 192 is 7^{th} term of the G.P.

**Example 2**: 5

^{th}term and 3

^{rd}term of a G.P is 256 and 16 respectively. Find its 8

^{th}term.

**Solution**:

^{4}= 256 —(1)

^{2}= 16—(2)

Dividing (1) by (2) gives,

r^{2} = 16

r = 4

Substituting r = 4 in (2) gives,

**Example 3**: Find the sum of the first 6 terms of the G.P 4, 12, 36,…..

**Solution**:

First term, a = 4

Common ratio, r = 12/4 = 3

n = 6

Sum of n terms of a G.P,

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