Question

Prove that for all positive integers n, 1 4 ··· n2 =n(n 1)(2n 1)/6= 13n3 21n2 61n

Answer 1

To prove the equation 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6 = 13n^3 - 21n^2 + 61n, we can use mathematical induction.

Step-by-step explanation:

To prove that 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6 = 13n^3 - 21n^2 + 61n, we can use the method of mathematical induction. We start with the base case, n = 1, and verify that the equation holds. Then we assume that the equation holds for some positive integer k, and we prove that it also holds for k+1. By doing so, we establish that the equation holds for all positive integers n using mathematical induction.