Pascal's Triangle and Binomial Coefficients

Question

Prove that if you move straight down in Pascal's Triangle, visiting every other row, then the numbers are increasing.

Answer 1

...prove that $\frac{n!}{2(n/2)!}$ for $2 | n$ is increasing as $n$ increases? Or down starting at arbitrary $k$ ans $n$? The latter is also fairly trivial:

$\binom{n+2}{k+1} = \binom{n+1}{k} + \binom{n+1}{k+1} = \bigg(\binom{n}{k-1} + \binom{n}{k}\bigg) + \binom{n+1}{k+1} > \binom{n}{k} $