What's happening in this summation? $x^{n+1} + \sum_{k=1}^n \binom {n+1}{k} x^{n-k+1}y^k + y^{n+1} = \sum_{k=0}^{n+1} \binom {n+1}{k} x^{n+1-k}y^k$

Question

Going through a lecture I encountered this fragment of one of induction proofs.

$$x^{n+1} + \sum_{k=1}^n \binom {n+1}{k} x^{n-k+1}y^k + y^{n+1} = \sum_{k=0}^{n+1} \binom {n+1}{k} x^{n+1-k}y^k$$

And as I'm not quite familiar yet with summation I'm really struggling to figure out why this is equal. Any help and explanation would be greatly appreciated.

The terms $x^{n+1}$ and $y^{n+1}$ correspond to the summands in the right-hand side obtained for $n=0$ and $n=k+1$. Notice these index are not taken in the first summation. — Federico T., Sep 03 '23 at 10:49

score 1 · Answer 1 · answered Sep 03 '23 at 10:51

1

\begin{align} &LHS-RHS\\&=x^{n+1}+y^{n+1}-\binom {n+1}{0}x^{n+1}y^0-\binom {n+1}{n+1}x^{0}y^{n+1}\\&=x^{n+1}+y^{n+1}-x^{n+1}-y^{n+1}\\&=0 \end{align}

answered Sep 03 '23 at 10:51

Ricky

3,148

What's happening in this summation? $x^{n+1} + \sum_{k=1}^n \binom {n+1}{k} x^{n-k+1}y^k + y^{n+1} = \sum_{k=0}^{n+1} \binom {n+1}{k} x^{n+1-k}y^k$

1 Answers1