Answer
See the explanation below.
Work Step by Step
${{S}_{1}}$: 2 is a factor of ${{1}^{2}}+5\left( 1 \right)=6$ since $6=2\cdot 3$
${{S}_{k}}$: 2 is a factor of ${{k}^{2}}+5k $
${{S}_{k+1}}$: 2 is a factor of ${{\left( k+1 \right)}^{2}}+5\left( k+1 \right)$
$\begin{align}
& {{\left( k+1 \right)}^{2}}+5\left( k+1 \right)={{k}^{2}}+2k+1+5k+5 \\
& ={{k}^{2}}+7k+6 \\
& ={{k}^{2}}+5k+2\left( k+3 \right) \\
& =\left( {{k}^{2}}+5k \right)+2\left( k+3 \right)
\end{align}$
Since, we assume ${{S}_{k}}$ is true, then, we know 2 is a factor of ${{k}^{2}}+5k $. Since 2 is a factor of $2\left( k+3 \right)$, we conclude 2 is a factor of the sum $\left( {{k}^{2}}+5k \right)+2\left( k+3 \right)$.
If ${{S}_{k}}$ is true, then ${{S}_{k+1}}$ is also true.
