Before diving deep into finding the Sum of Squares of First n Natural Numbers and Infinite GP, let’s first understand Geometric Progression. Geometric Progression (GP) is a sort of mathematical sequence in which each succeeding term is produced by multiplying each preceding term by a set integer known as a common ratio. This progression is also known as a pattern-following geometric sequence of numbers. The common ratio multiplied by each term to get the next term is not zero. A Geometric series would be 2, 4, 8, 16, 32, 64,…, where the common ratio is 2. Now, let’s turn back to our topic and learn about how to find the **Sum of Squares of First n Natural Numbers** and Infinite GP in detail.

**Sum of Infinite GP**

The **Sum of Infinite GP** is simply the sum of a GP’s infinite terms (Geometric Progression). A limited or infinite GP can exist. The formula for calculating the sum of an infinite GP’s first ‘n’ terms is Sn = a(1 – rn) / (1 – r), where ‘a’ is the first term and ‘r’ is the GP’s common ratio.

**Sum of Infinite GP When |r| is Less Than 1**

We’ve already seen how to compute the sum of GP’s infinite terms. Let us create a formula for this. Consider a Geometric Progression whose first term is ‘a’ and whose common ratio is ‘r,’ where |r| = 1. The sum of its infinite terms is then:

S = a + ar + ar2 + ar3 + … (i)

Multiply both sides by ‘r’:

rS = ar + ar2 + ar3 + …(ii)

Subtracting (ii) from (i):

S – rS = a

S (1 – r) = a

Dividing the two sides by (1 – r),

S = a / (1 – r)

When r is smaller than one, this is the formula for infinite GP.

**Sum of Infinite GP When |r| is Equal and Greater Than 1**

As we saw in the first section, the sum of infinite GP is a finite number since the terms of GP are essentially equal to zero after a finite number of terms when |r| = 1. When |r| ≥ 1, the GP is something like 2, 4, 6, 8, 16, 32, 64, 128, 256, 512, 1024,…. where the numbers keep increasing and eventually become very huge numbers. As a result, the total of a GP where |r| is bigger than 1 cannot be calculated, and the series diverges in this instance.

**Sum of Infinite GP Formulas**

We may summaries the sum of Infinite GP formulae as follows:

S∞ = a / (1 – r) when |r| < 1

S∞ = ±∞, when |r| ≥ 1

where ‘a’ is the first term and ‘r’ is the common ratio.

**Sum of Squares of n Natural Numbers**

Let us first recall the definition of natural numbers. Natural numbers are the counting numbers ranging from 1 to infinity. If we consider n consecutive natural numbers, the sum of their squares is represented as n2, where n runs from 1 to infinity.

You can calculate the sum of squares of the first n natural numbers by using the formula,

Sn = 12 + 22 + 32 + … + n2 = [n(n+1)(2n+1)] / 6.

This formula can be proved using the principle of mathematical induction.

**Sum of Squares of n Natural Numbers Formula**

The formulas for calculating the sum of squares of n natural numbers, the sum of squares of the first n even numbers, and the sum of squares of the first n odd numbers are as follows:

Sum of squares of n natural numbers=[n(n+1)(2n+1)] /6

Sum of squares of first n odd numbers=[n(2n+1)(2n-1)] / 3

Sum of squares of first n even numbers=[2n(n + 1)(2n + 1)] / 3

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