Answer
The required value of ${{S}_{1}}=0$, ${{S}_{2}}=2$ and ${{S}_{3}}=6$, and also the statement is true.
Work Step by Step
Let us consider the statement:
${{S}_{n}}:2\text{ is a factor of }{{n}^{2}}-n $.
For ${{S}_{1}}$ one has
$\begin{align}
& {{S}_{1}}:{{n}^{2}}-n={{1}^{2}}-1 \\
& =0
\end{align}$
Therefore, the above statement is true for $ n=1$ as 2 is a factor of 0.
For ${{S}_{2}}$ one has
$\begin{align}
& {{S}_{2}}:{{2}^{2}}-2=4-2 \\
& =2
\end{align}$
Now, the above statement is true for $ n=2$ as 2 is a factor o f2.
For ${{S}_{3}}$ one has
$\begin{align}
& {{S}_{3}}:{{3}^{2}}-3=9-3 \\
& =6
\end{align}$
Thus, the above statement is true for $ n=3$ as 2 is a factor of 6.
The values are ${{S}_{1}}=0,\,\,{{S}_{2}}=2$, and ${{S}_{3}}=6$. Also, the statement is true.
