You have taken, .

This is incorrect (substitute n=2 and the right hand side becomes undefined).

Much easier is to treat the summation without breaking it apart into two pieces.

You have to evaluate,

When r=1;

When r=2;

When r=3;

.

.

.

.

.

.

When r=n-1;

When r=n;

Adding all these together you will get the correct answer.