Let $X$ be a binomial random variable with parameters $n$ and $p$.
How do I show the following? $P(X=k+1)=\frac{p}{1-p}\frac{n-k}{k+1}P(X=k), {\ }k=0,1,...,n-1$ As $k$ goes from $0$ to $n$, $P(X=k)$ first increases and then decreases. How do I show that this probability reaches its largest value when $k$ is the largest integer less than or equal to $(n+1)p$?