Condition for Increasing Binomial Coefficients/Proof 1

Theorem

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

Let $\dbinom n k$ denote a binomial coefficient for $k \in \N$.

Then:

$\dbinom n k < \dbinom n {k + 1} \iff 0 \le k < \dfrac {n - 1} 2$

Proof

\(\ds \dbinom n k\)

\(<\)

\(\ds \dbinom n {k + 1}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \frac {n!} {\paren {n - k}! k!}\)

\(<\)

\(\ds \frac {n!} {\paren {n - k - 1}! \paren {k + 1}!}\)

Definition of Binomial Coefficient

\(\ds \leadstoandfrom \ \ \)

\(\ds k + 1\)

\(<\)

\(\ds n - k\)

Multiplying both sides by $\dfrac {\paren {n - k}! \paren {k + 1}!} {n!}$

\(\ds \leadstoandfrom \ \ \)

\(\ds 2 k\)

\(<\)

\(\ds n - 1\)

\(\ds \leadstoandfrom \ \ \)

\(\ds k\)

\(<\)

\(\ds \frac {n - 1} 2\)

$\blacksquare$