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Or must I prove this via induction on $n$?

My idea is to simply use the identity $\binom{n + 1}{k} = \binom{n}{k - 1} + \binom{n}{k}$ as follows

$$\sum\limits_{i = 0}^{k} \binom{n + i}{i} = \binom{n + k + 1}{k}$$ $$\sum\limits_{i = 0}^{k + 1} \binom {n + i}{i} = \binom{n + k + 1}{k} + \binom{n + k + 1}{k + 1}$$ $$\sum\limits_{i = 0}^{k + 1} \binom {n + i}{i} = \binom{n + k + 2}{k + 1}$$

but I'm not sure if this is valid and I'd like an explanation for why I should choose one over the other.

mooglin
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  • You shouldn't write the thing you are proving at the start. Also, you need to prove a base case for induction to work here. – pancini Aug 07 '20 at 22:26
  • As for the question of if you should do induction over $n$ or induction over $k$, the answer is that you should be doing double-induction over both – JMoravitz Aug 07 '20 at 22:29
  • What exactly is double-induction? I can't find any mention of this on the internet – mooglin Aug 09 '20 at 22:42

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