#### Answer

The required the solution is ${{S}_{k}}:2$ is a factor of ${{k}^{2}}-k $, ${{S}_{k+1}}:2$ is a factor of ${{k}^{2}}+k $.

#### Work Step by Step

Then, we are using ${{S}_{k}},{{S}_{k+1}}$ from the expression ${{S}_{n}}:2$ is a factor of ${{n}^{2}}-n $.
${{S}_{k}}:2$ is a factor of ${{k}^{2}}-k $
${{S}_{k+1}}:2$ is a factor of ${{\left( k+1 \right)}^{2}}-\left( k+1 \right)$
After solving,
$\begin{align}
& {{\left( k+1 \right)}^{2}}-\left( k+1 \right)={{k}^{2}}+2k+1-k-1 \\
& ={{k}^{2}}+k
\end{align}$
Thus, the solution is, ${{S}_{k}}:2$ is a factor of ${{k}^{2}}-k $, ${{S}_{k+1}}:2$ is a factor of ${{k}^{2}}+k $.