Prove this binomial coefficient equation

Question

I'm supposed to prove the following: $$ \sum_{i=0}^{n-1} \binom{2n-2-i}{i} = \sum_{i=0}^{n-1} \binom{n-1+i}{i}$$

Is there any simple conversion to come from the first term to the second one?

Answer 1

This is not true. Say $n=2$, we get:

$$ \sum_{i=0}^1 \binom{2-i}{i} = {2\choose 0}+ {1\choose 1} = 1+1=2$$

but $$\sum_{i=0}^1 \binom{1+i}{i}= {1\choose 0}+ {2\choose 1}= 1+2=3 $$