Expression using powers of n for recursive summation notation

Question

I have this question:

Find an explicit expression, i.e. a simple fraction involving powers of n, for the following sum:

$\sum_{j=1}^n \sum_{k=j}^n k$

Setting n at 5, I can see that the notation gives the sum of numbers j to n on each run through, so like this:

j=1 gives $(1+2+3+4+5)$

j=2 gives $(2+3+4+5)$

j=3 gives $(3+4+5)$

j=4 gives $(4+5)$

j=5 gives $(5)$

However, this is as far as I can get before hitting a brick wall. I can't see how you'd get an expression involving powers of n to perform that same summation. Any help is VERY much appreciated; I'm remarkably new to this area of mathematics.

Answer 1

If you do the outer sum for the first j terms you spell out for $n=5$, you get

$1 + 2 + 2 + 3 + 3 + 3 \ldots$

$1 + 2 \times 2 + 3 \times 3 + 4 \times 4 + 5 \times 5$

which can be expressed as

$\sum_{j = 1}^{n} j^{2}$

Answer 2

Hint: $1+2+3+...+n = \frac{n(n+1)}{2}$