Answer
The value of ${{S}_{1}}=0$, ${{S}_{2}}=6$ and ${{S}_{3}}=24$, and also the statement is true.
Work Step by Step
Let us consider the statement:
${{S}_{n}}:3\text{ is a factor of }{{n}^{3}}-n $
For ${{S}_{1}}$ one has
${{1}^{3}}-1=0$
So, the above statement is true for $ n=1$ as 3 is a factor of 0.
For ${{S}_{2}}$ one has
$\begin{align}
& {{2}^{3}}-2=8-2 \\
& =6
\end{align}$
Therefore, the above statement is true for $ n=2$ as 3 is a factor of 6.
For ${{S}_{3}}$ one has
$\begin{align}
& {{3}^{3}}-3=27-3 \\
& =24
\end{align}$
Thus, the above statement is true for $ n=3$ as 3 is a factor of 24.
The values are ${{S}_{1}}=0,{{S}_{2}}=6$, and ${{S}_{3}}=24$. Also, the statement holds true.
