\begin{eqnarray}
\nonumber\displaystyle (1+1)^{k+1}=\sum_{i=0}^{k+1}{k+1 \choose i}1^i\\
\nonumber\displaystyle (1+2)^{k+1}=\sum_{i=0}^{k+1}{k+1 \choose i}2^i\\
\nonumber\vdots\\
\nonumber\displaystyle (1+n)^{k+1}=\sum_{i=0}^{k+1}{k+1 \choose i}n^i
\end{eqnarray}