It wasn't that hard to do, I just can't remember the method.
The TPH solution is correct and complete!
But letís say we need to do it without calculus.
We know that n is odd. Suppose that the are two values b & c such that
Note that .
So if k is even then n-k+1 is even so that is a positive term. Moreover, if k is odd then is also a positive term. Thus the only way is that , in other words, b is a unique root.