Comments on $\sum_{k=0}^{n} \binom{n}{k}=2^{n} \overset{?}{\iff} \sum_{k=0}^{n} \binom{2n+1}{k}=2^{2n}$
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$\sum_{k=0}^{n} \binom{n}{k}=2^{n} \overset{?}{\iff} \sum_{k=0}^{n} \binom{2n+1}{k}=2^{2n}$
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Jack D'Aurizio narratively proved $\color{red}{\sum\limits_{k=0}^{n} \binom{2n+1}{k}=2^{2n}}$. Is this red equation related, and can it be transmogrified, to $\color{limegreen}{\sum\limits_{k=0}^{n} \binom{n}{k}=2^{n}}$?
I started my attempt by substituting $n = m/2$, because the RHS of the green target equation has the form $\color{limegreen}{2^?}$. Then $\color{red}{\sum\limits_{k=0}^{n} \binom{2n+1}{k}=2^{2n} \implies \sum\limits_{k=0}^{m/2} \binom{m+1}{k}=2^{m}}$. But now what do I do? I can't rewrite $m$, again because the RHS of the target equation has the form $\color{limegreen}{2^?}$.
- Give a story proof that $\sum\limits_{k=0}^{n} \binom{n}{k}=2^{n}$.
Blitzstein. Introduction to Probability (2019 2 ed). p 35.
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