(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

How can i prove this relationship

\sum _{i=0}^k \text{Binomial}[n+1,k-2i] - \sum _{i=0}^k \text{Binomial}[n,k-2i]=\sum _{i=0}^k \text{Binomial}[n,k-1-2i]

2. Relevant equations

Binomial (n,k)=n^k/k!

3. The attempt at a solution

I attempted subbing into mathyematica but this didn't work so i attempted by hand and got completely lost. Any helpful comments would be helpful.

Result from Mathematica

-Binomial[n, -1 + k] HypergeometricPFQ[{1, 1/2 - k/2,

1 - k/2}, {1 - k/2 + n/2, 3/2 - k/2 + n/2}, 1] -

Binomial[n,

k] HypergeometricPFQ[{1, 1/2 - k/2, -(k/2)}, {1/2 - k/2 + n/2,

1 - k/2 + n/2}, 1] +

Binomial[1 + n,

k] HypergeometricPFQ[{1, 1/2 - k/2, -(k/2)}, {1 - k/2 + n/2,

3/2 - k/2 + n/2}, 1]

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# Homework Help: Summation with Binomial Expansion

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