Weak principle of induction for $5+10+15+\ldots+5n= \frac{5n(n+1)}{2}$ 
Show that
$$5+10+15+\ldots+5n= \frac{5n(n+1)}{2}$$

Proving the base case $n(1)$:
$5(1)= \frac{5(1)(1+1)}{2}$
$5 = \frac{5(2)}{2}$
$5 = 5$
Induction hypothesis:
$n = k$
$5+10+15+\ldots+5k = \frac{5k(k+1)}{2}$
Induction step (adding $k+1$):
$5+10+15+\ldots+5k+5k+1 = \frac{5k+1(k+1+1)}{2}$
Substituting $\frac{5k(k+1)}{2}$ for $5k$:
$\frac{5k(k+1)}{2}+5k+1 = \frac{(5k+1)(k+1+1)}{2}$
Simplifying:
$\frac{5k(k+1)+2(5k+1)}{2} = \frac{(5k+1)(k+2)}{2}$
$\frac{5k^2+5k+10k+2}{2} = \frac{5k^2+10k+k+2}{2}$
These aren't equal, so what did I do wrong here?

When you plug in $n=k+1$ you have that
$$5+10+\cdots+5k+5(k+1)\not=5+10+\cdots+5k+5k+1$$
