Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index
Theorem
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 $S_n = \sequence {x_k}$ be the sequence defined as:
- $x_k = \dbinom n k$
Then $S_n$ is strictly increasing exactly where $0 \le k < \dfrac n 2$.
Corollary 1
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 $\dfrac n 2 < k \le n$.
Corollary 2
Maximum Value of Binomial Coefficient
Proof
When $k \ge 0$, we have:
| \(\ds \binom n {k + 1}\) | \(=\) | \(\ds \frac {n!} {\paren {k + 1}! \paren {n - k - 1}!}\) | Definition of Binomial Coefficient | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {n - k} {n - k} \frac {n!} {\paren {k + 1}! \paren {n - k - 1}!}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {n - k} {\paren {k + 1} \paren {n - k} } \frac {n!} {k! \paren {n - k - 1}!}\) | extracting $k + 1$ from its factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {n - k} {k + 1} \frac {n!} {k! \paren {n - k}!}\) | inserting $n - k$ into its factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {n - k} {k + 1} \binom n k\) | Definition of Binomial Coefficient |
In order for $S_n$ to be strictly increasing, it is necessary for $\dfrac {n - k} {k + 1} > 1$.
So:
| \(\ds \dfrac {n - k} {k + 1}\) | \(>\) | \(\ds 1\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds n - k\) | \(>\) | \(\ds k + 1\) | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds n\) | \(>\) | \(\ds 2 k + 1\) | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds n\) | \(>\) | \(\ds 2 \paren {k + 1} - 1\) |
Thus $\dbinom n {k + 1} > \dbinom n k$ if and only if $k + 1$ is less than half of $n$.
Hence the result.
$\blacksquare$