How to find arithmetic mean of number

Question

The arithmetic mean of 1,8,27,64... upto n terms is given by .... I know the formula for arithmetic mean But i don't know how to apply it I applied a+b/2 But it's not logical here so Plzz tell me the exact solution of This question

Ans is n (n+1)^2/4

Plzz explain me in a detail

Answer 1

Your numbers are perfect cubes.

So, you want $$\frac{\sum_{i=1}^n i^3}n$$ Use Faulhaber's formulae.

Answer 2

$$(n+1)^4-1=\sum_{k=1}^n((k+1)^4-k^4)=$$ $$=\sum_{k=1}^n(4k^3+6k^2+4k+1)=4\sum_{k=1}^nk^3+n(n+1)(2n+1)+2n(n+1)+n.$$ Can you end it now?