Sequence of Binomial Coefficients is Strictly Decreasing from Half Upper Index

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Corollary to Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index

Let $n \in \Z_{>0}$ be a strictly positive integer.

Let $\dbinom n k$ be the binomial coefficient of $n$ over $k$ for a positive integer $k \in \Z_{\ge 0}$.

Let $S_n = \left\langle{x_k}\right\rangle$ be the sequence defined as:

$x_k = \dbinom n k$


Then $S_n$ is strictly decreasing exactly where $\dfrac n 2 < k \le n$.


Proof

If $k > \dfrac n 2$ then it follows that $n - k < \dfrac n 2$.


Thus:

\(\displaystyle \binom n {k + 1}\) \(=\) \(\displaystyle \binom n {n - \left({k + 1}\right)}\) Symmetry Rule for Binomial Coefficients
\(\displaystyle \) \(=\) \(\displaystyle \binom n {n - k - 1}\)
\(\displaystyle \) \(<\) \(\displaystyle \binom n {n - k}\) Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index
\(\displaystyle \) \(=\) \(\displaystyle \binom n k\) Symmetry Rule for Binomial Coefficients

Hence the result.

$\blacksquare$