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 $n=1$:
$3n(n+1)=3(1)(1+1)=3(2)=6$
It is correct!
Now, suppose that the formula is correct, that is:
$6+12+18+24+...+6n=3n(n+1)$
Now, let's prove the formula for $n+1$:
$6+12+18+24+...+6n+6(n+1)=(6+12+18+24+...+6n)+6(n+1)=3n(n+1)+6(n+1)=(3n+6)(n+1)=3(n+2)(n+1)=3(n+1)(n+2)=3(n+1)[(n+1)+1]$
That is exactly the given formula if $n$ is changed by $n+1$.
