Answer
The formula was proved for $n=1$
The formula is correct if $n$ is changed by $n+1$
Work Step by Step
Let's prove the formula for $n=1$:
$\frac{1(1+1)}{2}=\frac{1(2)}{2}=1$
It is correct!
Now, suppose that the formula is correct, that is:
$1+2+3+4+...+n=\frac{n(n+1)}{2}$
Now, let's prove the formula for $n+1$:
$1+2+3+4+...+n+(n+1)=(1+2+3+4+...+n)+(n+1)=\frac{n(n+1)}{2}+(n+1)=\frac{n(n+1)}{2}+\frac{2(n+1)}{2}=\frac{(n+1)(n+2)}{2}=\frac{(n+1)[(n+1)+1]}{2}$
That is exactly the given formula if $n$ is changed by $n+1$
