Suppose that X and Y are independent discrete random variables with the same probability mass function

p_{X}(k) = p_{Y}(k) =pq^(k-1)

where 0<p<1. Show that for any integer n greater than or equal to 2, the conditional probability mass function

p(k) = P(X=k|X+Y=n)

is a discrete uniform distribution

So I do so in this way

P(X=k|X+Y=n)

=P(X=k|Y=n-k)

=P(X=k, Y=n-k)/P(Y=n-k)

=$\displaystyle p(k,n-k)$/P(Y=n-k)

=p^{2}q^{n-1}+p^{2}q^{n-1}+..../pq^{n-k}

=np^{2}q^{n-1}/pq^{n-k}

=nq^{k-1}

which should be wrong in my opinion. So what should be the correct ans?