Putnam 2020 A2

For $k$ a non-negative integer, evaluate
\begin{align*}
\sum_{j=0}^k 2^{k-j} \binom{k +j}{j}.
\end{align*}

Apparently, I only know how to do induction anymore now but this problem is straightforward using it. We suppose the inductive hypothesis, and assume that for a fixed $k$ that
\begin{align*}
\sum_{j=0}^k 2^{k-j} \binom{k +j}{j} = 2^{2k}.
\end{align*}
We will proceed to show
\begin{align*}
\sum_{j=0}^{k + 1} 2^{k + 1-j} \binom{k + 1 +j}{j} = 2^{2k + 2}.
\end{align*}

By a simple binomial identity, we have
\begin{align*}
\sum_{j=0}^{k + 1} 2^{k + 1-j} \binom{k + 1 +j}{j} &= \sum_{j=0}^{k + 1} 2^{k + 1-j} \binom{k+j}{j} + \sum_{j=0}^{k + 1} 2^{k + 1-j} \binom{k +j}{j – 1}.
\end{align*}
Let $A$, $B$ be the first and second term respectively on the right hand side, then
\begin{align*}
A &= 2\sum_{j=0}^{k} 2^{k-j} \binom{k+j}{j} + \binom{2k + 1}{k+1} \\
&= 2^{2k + 1} + \binom{2k + 1}{k+1}.
\end{align*}
For the other term,
\begin{align*}
B &= \sum_{j=1}^{k + 1} 2^{k + 1-j} \binom{k +j}{j – 1} \\
&= \sum_{j=0}^{k} 2^{k-j} \binom{k +1 +j}{j} \\
&= \frac{1}{2} \sum_{j=0}^{k + 1} 2^{k + 1-j} \binom{k +1 +j}{j} – \frac{1}{2} \binom{2k + 2}{k + 1}.
\end{align*}
Thus, we rearranging, we have our result
\begin{align*}
\frac{1}{2}\sum_{j=0}^{k + 1} 2^{k + 1-j} \binom{k + 1 +j}{j} &= 2^{2k+1} + \binom{2k + 1}{k+1} – \frac{1}{2} \binom{2k + 2}{k + 1} \\
&= 2^{2k+1}.
\end{align*}

Leave a Reply

Your email address will not be published.

This site uses Akismet to reduce spam. Learn how your comment data is processed.