Poisson distribution, showing the mode of $X$

Question

I need help solving part b of this question:

The random variable $X \sim ~Po(m)$.

Given that $P(X=k+1)=P(X=k-1)$, where $k$ is a positive integer,

a) show that $m^2=k(k+1)$

b) hence show that the mode of $X$ is $k$.