Proof of the Sum of the Series ∑r²

Definition of Sum of Series ∑r²

The series ∑r² is defined as the summation of the squares of each term of an arithmetic series where ‘r’ is the term number.
In simple terms, ∑r² = 1² + 2² + 3² + … + n² for a series of ‘n’ terms.

The formula for the sum of the series ∑r² is given by: n(n+1)(2n+1)/6.
Use this formula to calculate the sum of the squares of the first ‘n’ natural numbers.

To prove this formula, we can use the method of mathematical induction, which involves two steps: the base case and the induction step.
The base case is to show that the formula holds for n = 1. Plugging 1 into the formula, we get 1(1+1)(2*1+1)/6 = 1.
The induction step involves assuming the formula holds for n = k and showing it must then hold for n = k + 1.
Assuming the formula holds for n = k, we get the sum of the squares of the first k natural numbers as k(k+1)(2k+1)/6.
We then add the square of k+1 to both sides: k(k+1)(2k+1)/6 + (k+1)².
With some algebraic manipulation, the right-hand side becomes: (k+1)(k+2)(2k+3)/6. This is the same as the original formula with n replaced by k+1. Hence, the formula holds for n = k + 1.
Thus, the formula is proven by mathematical induction, giving a method for calculating the sum of the series ∑r².

The summation of squares series has several applications. This formula is used in various calculations involving sequence and series, probability, and statistics.
It is also crucial for understanding and solving problems involving quadratic sequences or other sequences involving squares of terms.