Pascal's Triangle

Question

My question is the following;

Q: Prove that if we move straight down in Pascal’s Triangle (visiting every other row), then the numbers we see are increasing.

Found an answer but that doesn't count as a proof I think. By the way I couldn't write down what I found because I don't know LaTeX forgive me for that.

So, could you help me please?

Answer 1

Basically, you're comparing $\binom{n}{k}$ and $\binom{n+2}{k+1}$. Note that \begin{align*} \frac{\binom{n+2}{k+1}}{\binom{n}{k}} = \frac{(n+1)(n+2)}{(n-k+1)(k+1)} >1, \end{align*} and so, you're done.