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