Can someone please show why the following holds:
$$ \sum_{k=0}^n\binom{n-1}{k-1}p^{k}q^{n-k} = p \sum_{k=1}^n\binom{n-1}{k-1}p^{k-1}q^{n-k} $$
Thanks in advance.
Edit:
Thank you for commenting.
This is from Blitzstein and Hwang, p143.
The LHS can be transformed to:
$$ \sum_{k=0}^n\binom{n-1}{k-1}p^{k}q^{n-k} = \binom{n-1}{-1}q^{n-k} + p\sum_{k=1}^n\binom{n-1}{k-1}p^{k-1}q^{n-k} $$
The identity holds if $$ \binom{n-1}{-1}q^{n-k} = 0. $$