Answer
See below.
Work Step by Step
Using the hint $2S=[1+n]+[2+(n-1)]+...+[n+1]=[1+n]+[1+n)]+...+[n+1]=n(n+1)$ (we have $n$ terms originally (because the left number of the summation goes from $1$ to $n$ with increments of $1$) and each term sums to $n+1$.)
Thus $S=\frac{n(n+1)}{2}$.
Thus we proved what we had to.