Answer
The value of ${{S}_{1}}=1$, ${{S}_{2}}=4$ and ${{S}_{3}}=9$, and also the statement is true.
Work Step by Step
We consider the statement:
${{S}_{n}}:1+3+5+\cdots +\left( 2n-1 \right)={{n}^{2}}$
For ${{S}_{1}}$ we has
${{S}_{1}}:1={{1}^{2}}$
Therefore, the above statement is true for $ n=1$.
For ${{S}_{2}}$ one has
${{S}_{2}}:1+3={{2}^{2}}$
Therefore, the above statement is true for $ n=2$.
For ${{S}_{3}}$ one has
${{S}_{3}}:1+3+5={{3}^{2}}$
Therefore, the above statement is true for $ n=3$.
Thus, the statement holds.
Thus the values are ${{S}_{1}}=1,\,\,{{S}_{2}}=4$, and ${{S}_{3}}=9$. And the statement is true.
