If Sk = (1 + 2 + 3 + ....)/k, prove that, (S1^2 + S2^2 + .... Sn^2) = n/24(2n^2 + 9n + 13)​​​​​​​.

Question

If \(S_k = \frac{(1+2+3+....k)}{k},\) prove that, (S₁² + S₂² + .... S_n²) = n/24(2n² + 9n + 13).

S_k = (1 + 2 + 3 + ....)/k

Answer 1

Given, S_k = (1 + 2 + 3 + ....)/k

To prove:

Note:

Now, the Left hand side of the condition given in the question can be written as,

So,