Answer
See the explanation below.
Work Step by Step
We use mathematical induction as follows:
Statement ${{S}_{1}}$ is:
${{S}_{1}}=3$
Then, simplifying on the right, obtain
$\begin{align}
& \frac{n\left( n+5 \right)}{2}=\frac{\left( 1 \right)\left( 1+5 \right)}{2} \\
& =3
\end{align}$.
This statement shows that ${{S}_{1}}$ is true,
Suppose ${{S}_{k}}$ is true. Using ${{S}_{k}},{{S}_{k+1}}$ from the expression,
${{S}_{k}}=3+4+5+....+\left( k+2 \right)=\frac{k\left( k+5 \right)}{2}$
Adding $\left( k+3 \right)$ on both sides as:
$\begin{align}
& 3+4+5+....+\left( k+3 \right)=\frac{\left( k \right)\left( k+5 \right)}{2}+k+3 \\
& =\frac{{{k}^{2}}+7k+6}{2} \\
& =\frac{\left( k+1 \right)\left( k+6 \right)}{2} \\
& =\frac{\left( k+1 \right)\left( \left( k+1 \right)+5 \right)}{2}
\end{align}$
Thus,
${{S}_{k+1}}$ is true. The result, ${{S}_{n}}=3+4+5+....+\left( n+2 \right)=\frac{n\left( n+5 \right)}{2}$ is true by mathematical induction.
Hence, the provided statement is proved by mathematical induction.
