Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.1 Sequences - 11.1 Assess Your Understanding - Page 827: 52

$4+9+16+25+36+...+(n+1)^{2}$

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 terms as follows: $\displaystyle \sum_{k=1}^{n}(k+1)^{2}=(1+1)^{2}+(2+1)^{2}+(3+1)^{2}+..+(n+1)^{2} \\ =4+9+16+25+36+...+(n+1)^{2}$