Question

induction: how do you know when to use 1 or 0 for the basis step? For example \[\left(\begin{matrix}n\\ 0\end{matrix}\right)+\left(\begin{matrix}n+1 \\ 1\end{matrix}\right)+...+\left(\begin{matrix}n+r \\ r\end{matrix}\right)=\left(\begin{matrix}n+r+1 \\ r\end{matrix}\right)\]

Answer 1

I don't see how letting r=0 verifies the equation because don't you get
\[\left(\begin{matrix}n \\ 0\end{matrix}\right)=\left(\begin{matrix}n+0+1 \\ 0\end{matrix}\right)\]

Answer 2

Depends on which domain of n you want to prove it works for.