Talk:Hockey-stick identity

Alternative proof

Here is another proof:

By the formula for the sum of a geometric series, ${\frac {(1+r)^{n+1}-1}{(1+r)-1}}=\displaystyle \sum _{m=0}^{n}(1+r)^{m}$

Now expand both sides via the binomial theorem and simplify:

$\displaystyle \sum _{m=0}^{n}{\binom {n+1}{m+1}}r^{m}=\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{o=0}^{m}{\binom {m}{o}}r^{o}$

The hockey stick identity follows by equating coefficients of $r^{k}$ .

I came up with this proof, which I think is pretty nice, and I can't find it anywhere else, so I just assume its new. EZ132 (talk) 19:09, 18 September 2020 (UTC)[reply]

Induction: does it require induction on both N and K?

As I understand it, you prove it works for some N and then show this works for N+1. But what about K? Does the proof in this article address different values of K or is that not needed? 50.230.251.244 (talk) 21:29, 4 November 2023 (UTC)[reply]