Give exact value of s as a function of n in closed form

Question

here is what I did so far but I cannot go further

Answer 1

The innermost step is very easy: $\sum_{k=1}^j 1$ just counts the integers from $1$ to $j$. For the next two steps you may find the following formulas useful: $$ \eqalign{\sum_{j=1}^n j &= n(n+1)/2\cr \sum_{j=1}^n j^2 &= n(n+1)(2n+1)/6\cr}$$