Answer
See the explanation below.
Work Step by Step
$\begin{align}
& {{S}_{1}}:5=\frac{5\left( 1 \right)\left( 1+1 \right)}{2} \\
& 5=\frac{5\left( 2 \right)}{2} \\
& 5=5\ \text{is}\ \text{true}
\end{align}$
$\begin{align}
& {{S}_{k}}=5+10+15+....+5k=\frac{5k\left( k+1 \right)}{2} \\
& {{S}_{k+1}}=5+10+15+....+5k+5\left( k+1 \right) \\
& {{S}_{k+1}}=\frac{5\left( k+1 \right)\left( k+2 \right)}{2}
\end{align}$
And add $5\left( k+1 \right)$ to both sides of ${{S}_{k}}$
$5+10+15+....+5k+5\left( k+1 \right)=\frac{5k\left( k+1 \right)}{2}+5k\left( k+1 \right)$
Then, simplify the right-hand side:
$\begin{align}
& \frac{5k\left( k+1 \right)}{2}+5\left( k+1 \right)=\frac{5k\left( k+1 \right)+10\left( k+1 \right)}{2} \\
& \frac{5k\left( k+1 \right)}{2}+5\left( k+1 \right)=\frac{\left( 5k+10 \right)\left( k+1 \right)}{2} \\
& \frac{5k\left( k+1 \right)}{2}+5\left( k+1 \right)=\frac{5\left( k+1 \right)\left( k+2 \right)}{2} \\
\end{align}$
When ${{S}_{k}}$ is true then ${{S}_{k+1}}$ is true.
