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:
\(\ds \binom n {k + 1}\) | \(=\) | \(\ds \binom n {n - \left({k + 1}\right)}\) | Symmetry Rule for Binomial Coefficients | |||||||||||
\(\ds \) | \(=\) | \(\ds \binom n {n - k - 1}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \binom n {n - k}\) | Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index | |||||||||||
\(\ds \) | \(=\) | \(\ds \binom n k\) | Symmetry Rule for Binomial Coefficients |
Hence the result.
$\blacksquare$