Answer
The required value of ${{S}_{1}}=3$, ${{S}_{2}}=7$ and ${{S}_{3}}=12$, and also the statement is true.
Work Step by Step
Let us c consider the statement:
${{S}_{n}}:3+4+5+\cdots +\left( n+2 \right)=\frac{n\left( n+5 \right)}{2}$
For ${{S}_{1}}$ one has
$\begin{align}
& {{S}_{1}}:3=\frac{1\cdot \left( 1+5 \right)}{2} \\
& =3
\end{align}$
Therefore, the above statement is true for $ n=1$.
For ${{S}_{2}}$ one has
$\begin{align}
& {{S}_{2}}:3+4=\frac{2\cdot \left( 2+5 \right)}{2} \\
& =7
\end{align}$
So, the above statement is true for $ n=2$.
For ${{S}_{3}}$ one has
$\begin{align}
& {{S}_{3}}:3+4+5=\frac{3\cdot \left( 3+5 \right)}{2} \\
& =12
\end{align}$
Therefore, the above statement is true for $ n=3$.
Thus, the values we obtain are ${{S}_{1}}=3,\,\,{{S}_{2}}=7$, and ${{S}_{3}}=12$. And the statement is true.
