# Geometric Sum Formula

Before going learn the geometric sum formula, let us recall what is a geometric sequence. A geometric sequence is a sequence where every term has a constant ratio to its preceding term. A geometric sequence with the first term a and the common ratio r and has a finite number of terms is commonly represented as a, ar, ar^{2}, ..., ar^{n-1}. A geometric sum is the sum of the terms in the geometric sequence. The geometric sum formula is used to calculate the sum of the terms in the geometric sequence.

## What Is the Geometric Sum Formula?

The geometric sum formula is defined as the formula to calculate the sum of all the terms in the geometric sequence. There are two geometric sum formulas. One is used to find the sum of the first n terms of a geometric sequence whereas the other is used to find the sum of an infinite geometric sequence.

### Geometric Sum Formula

1. The geometric sum formula for finite terms is given as:

If r = 1, S_{n} = na

If |r| < 1 , \(S_{n}=\dfrac{a(1-r^{n})}{1-r}\)

If |r| > 1, \(S_{n}=\dfrac{a(r^{n}-1)}{r-1}\)

2. The geometric sum formula for infinite terms is given as:

If |r| < 1, \(S_\infty = \dfrac{a}{1-r}\)

If |r| > 1, the series does not converge and it has no sum.

Where

- a is the first term
- r is the common ratio
- n is the number of terms

## Derivation of Geometric Sum Formula

The sum of a geometric series S_{n}, with common ratio *r* is given by: \(\mathrm{S}_n = \displaystyle{\sum_{i=1}^{n}\,a_i}\) = \(a\left(\dfrac{1 - r^n}{1 - r}\right)\). We will use polynomial long division formula.

- The sum of first n terms of the Geometric progression is
- \(S_n\) =a + ar + ar
^{2}+ ar^{3}+ ... + ar^{n–2}+ ar^{n–1 }------------> (1) - Multiplying both sides by r, we get
- r\(S_n\) =ar + ar
^{2}+ ar^{3}+ ... + ar^{n–2}+ ar^{n–1 }+ ar^{n }------------> (2) - (2)−(1) gives r\(S_n\) -\(S_n\) = ar
^{n}^{ }- a - ⇒ \(S_n\)(r-1) = a(r
^{n}^{ }- 1) - Thus we derive at the sum of n terms of GP as \(S_{n}=\dfrac{a(r^{n}-1)}{r-1}\)

Let us see the applications of the geometric sum formulas in the following section.

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## Examples Using Geometric Sum Formula

**Example 1:** Find the sum of the terms 1/3 + 1/9 + 1/27 + ... to ∞ using the geometric sum formula?

**Solution:**

To find: geometric sum

Given:

a = 1/3, r = 1/3

Using geometric sum formula for infinite terms,

S_{n} = a /(1-r)

S_{n} = (1/3)( 1 - 1/3)

S_{n} = 1/2

**Answer: Geometric sum of the given terms is 1/2.**

**Example 2: **Calculate the sum of series 1/5, 1/5, 1/5, .... if the series contains 34 terms.

**Solution:**

To find: geometric sum

Given:

a = 1/5, r = 1, and n = 34

Using geometric sum formula for finite terms,

S_{n} = na

S_{n} = 34 × 1/5

S_{n} = 6.8

**Answer:** Geometric sum of the given terms is 6.8.

**Example 3: **Find the sum of GP: 20, 60, 180, 540, and 1620, using the geometric sum formula.

Solution:

GP = 20, 60, 180, 540, and 1620 (given)

a = 20, r = 60/20 = 3, n = 5

Here r > 1.

Using geometric sum formula,

S_{n} = a[(r^{n}-1)/(r-1)]

S_{5} = 20[(3^{5}-1)/(3-1)]

= 20[(243-1)/2]

= 20[242/2]

=20 x 121

=2410

**Answer:** Geometric sum is 2410.

## FAQs on Geometric Sum Formula

### What Is the Geometric Sum Formula in Math?

In math, the geometric sum formula refers to the formula that is used to calculate the sum of all the terms in the geometric sequence. The two geometric sum formulas are:

- The geometric sum formula for finite terms: If r = 1, S
_{n}= an and if r≠1,S_{n}=a(1−r^{n})/1−r - The geometric sum formula for infinite terms: S
_{n}=a_{1}−r. If |r| < 1 , S_{∞ }= a/(1 - r)

### What Is the Geometric Sum Formula for Infinite Series?

In infinite series, there are two cases depending upon the value of r.

- Case 1: When |r| < 1, then S
_{∞}= a/(1 - r), where a is the first term and r is the common ratio - Case 2: |r| > 1, the series does not converge and it has no sum.

### What Is r in the Geometric Sum Formula for Finite Series?

In the geometric series formula, S_{n}=a(1−r^{n})/1−r, r refers to the common ratio in between the two consecutive terms.

### How To Use the Geometric Sum Formula?

For any given geometric series,

- Step 1: Check if it is a finite or an infinite series.
- Step 2: Identify the values of a (the first term), n (the number of terms), and r (the common ratio).
- Step 3: Put the values in an appropriate formula based on the common ratio.
- if r<1, sum = a(r
^{n}-1)/(r-1); if r>1, sum = a(1−r^{n})/1−r and if r = 1, sum = an

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