Solution
= Solution
For $k\ge n$, $p_k(n)=\binom{k-1}{n-1}p^n(1-p)^{k-n}$. It is the sum of $n$ independent geometric variables, so the <central limit theorem> applies with mean $n/p$ and standard deviation $\sqrt{n(1-p)}/p$. Take
$k_a(n)=\lceil n/p+a\sqrt{n(1-p)}/p\rceil$ and $k_b(n)=\lfloor n/p+b\sqrt{n(1-p)}/p\rfloor$.