Answer
$\displaystyle \frac{1}{2}+2+\frac{9}{2}+8+\frac{25}{2}+18+\frac{49}{2}+...+\frac{n^{2}}{2}$
Work Step by Step
We see that there are $n$ terms, as the index $k$ changes from $1$ to $n$. The index $k$ indicates how the terms are formed.
We write out the sum for the $n$ terms as follows:
$\displaystyle \sum_{k=1}^{n}\frac{k^{2}}{2}=\frac{(1)^{2}}{2}+\frac{(2)^{2}}{2}+\frac{(3)^{2}}{2}+...+\frac{(n)^{2}}{2} \\ =\displaystyle \frac{1}{2}+2+\frac{9}{2}+8+\frac{25}{2}+18+\frac{49}{2}+...+\frac{n^{2}}{2}$
