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)

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)