To prove the statement `1*2+2*3+3*4+cdots+n(n+1)=1/3n(n+2)(n+2)`  we need to use induction.

First, let n=1.

The left side is `1*2=2`

The right side is `1/3 1(2)(3)=2` so the statement is true for n=1.

Now assume `n=k` is true.

Then, we need to use that statement to show that `n=k+1` is also true, and by the principle of induction, means that the statement is true for all n.

For n=k+1, we need to show that

`1*2+2*3+3*4+cdots+n(n+1)+(n+1)(n+2)=1/3(n+1)(n+2)(n+3)`

`LS=1*2+2*3+3*4+cdots+n(n+1)+(n+1)(n+2)`  use induction for n=k

`=1/3n(n+1)(n+2)+(n+1)(n+2)`   simplify

`={n(n+1)(n+2)+3(n+1)(n+2)}/3`   

`=1/3(n+1)(n+2)(n+3)`

`=RS`

By induction, the statement is true.