# 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 $S_n = \left\langle{x_k}\right\rangle$ 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 $\dfrac n 2 < k \le n$.

## Proof

When $k \ge 0$, we have:

 $\displaystyle \binom n {k + 1}$ $=$ $\displaystyle \frac {n!} {\left({k + 1}\right)! \left({n - k - 1}\right)!}$ Definition of Binomial Coefficient $\displaystyle$ $=$ $\displaystyle \frac {n - k} {n - k} \frac {n!} {\left({k + 1}\right)! \left({n - k - 1}\right)!}$ $\displaystyle$ $=$ $\displaystyle \frac {n - k} {\left({k + 1}\right) \left({n - k}\right)} \frac {n!} {k! \left({n - k - 1}\right)!}$ extracting $k + 1$ from its factorial $\displaystyle$ $=$ $\displaystyle \frac {n - k} {k + 1} \frac {n!} {k! \left({n - k}\right)!}$ inserting $n - k$ into its factorial $\displaystyle$ $=$ $\displaystyle \frac {n - k} {k + 1} \binom n k$ Definition of Binomial Coefficient

In order for $S_n$ to be strictly increasing, it is necessary for $\dfrac {n - k} {k + 1} > 1$.

So:

 $\displaystyle \dfrac {n - k} {k + 1}$ $>$ $\displaystyle 1$ $\displaystyle \iff \ \$ $\displaystyle n - k$ $>$ $\displaystyle k + 1$ $\displaystyle \iff \ \$ $\displaystyle n$ $>$ $\displaystyle 2 k + 1$ $\displaystyle \iff \ \$ $\displaystyle n$ $>$ $\displaystyle 2 \left({k + 1}\right) - 1$

Thus $\dbinom n {k + 1} > \dbinom n k$ if and only if $k + 1$ is less than half of $n$.

Hence the result.

$\blacksquare$