What is wrong with this solution?(Headbang)

Problem:

Prove that for

Solution:

Proof by induction

Let the LHS =

RHS is

Is it valid for n=1?

LHS =

RHS= True!

Now let LHS =

Thats what we want our RHS to look like:

Now

Lets look att RHS.

Lets set (n+1) into the formula and subract n that should then equal the LHS hopefully:

.

.

.

.

RHS = LHS and by induction this is true for all n

Why is this not right?