Question 178
Let $r$ and $n$ be nonnegative integers such that $r \le n$.
a. Prove that
$$
\frac{n+1-2r}{n+1-r} \binom{n}{r}
$$
is an integer.
b. Prove that
$$
\sum_{r=0}^{\lfloor n/2 \rfloor} \frac{n+1-2r}{n+1-r} \binom{n}{r} \lt 2^{n-2}
$$
for all $n \ge 9$.
(Note that $\binom{n}{r} = \frac{n!}{r!(n-r)!}$. Also, if $x$ is a real number then $\lfloor x \rfloor$ is the unique integer such that $\lfloor x \rfloor \le x \lt \lfloor x \rfloor + 1$.)
a. Prove that
$$
\frac{n+1-2r}{n+1-r} \binom{n}{r}
$$
is an integer.
b. Prove that
$$
\sum_{r=0}^{\lfloor n/2 \rfloor} \frac{n+1-2r}{n+1-r} \binom{n}{r} \lt 2^{n-2}
$$
for all $n \ge 9$.
(Note that $\binom{n}{r} = \frac{n!}{r!(n-r)!}$. Also, if $x$ is a real number then $\lfloor x \rfloor$ is the unique integer such that $\lfloor x \rfloor \le x \lt \lfloor x \rfloor + 1$.)