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, ar², ..., 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 + ar² + ar³ + ... + ar^n–2 + ar^n–1 ------------> (1)
• Multiplying both sides by r, we get
• r\(S_n\) =ar + ar² + ar³ + ... + ar^n–2 + ar^n–1 + arⁿ  ------------> (2)
• (2)−(1) gives r\(S_n\)  -\(S_n\) = arⁿ - a
• ⇒ \(S_n\)(r-1) = a(rⁿ - 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.

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[(rⁿ-1)/(r-1)]

S₅ = 20[(3⁵-1)/(3-1)]

= 20[(243-1)/2]

= 20[242/2]

=20 x 121

=2410

Answer: Geometric sum is 2410.