# Chapter 10 - Section 10.4 - Mathematical Induction - Exercise Set - Page 1085: 10

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 ${{k}^{2}}+k$.

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