Uniqueness of Real z such that x Choose n+1 Equals y Choose n+1 Plus z Choose n

From ProofWiki
Jump to navigation Jump to search

Theorem

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

Let $x, y \in \R$ be real numbers which satisfy:

$n \le y \le x \le y + 1$


Then there exists a unique real number $z$ such that:

$\dbinom x {n + 1} = \dbinom y {n + 1} + \dbinom z n$

where $n - 1 \le z \le y$.


Proof

We have:

\(\ds \dbinom y {n + 1}\) \(\le\) \(\ds \dbinom x {n + 1}\) Ordering of Binomial Coefficients
\(\ds \) \(\le\) \(\ds \dbinom {y + 1} {n + 1}\)
\(\ds \) \(=\) \(\ds \dbinom y {n + 1} + \dbinom y n\) Pascal's Rule




Sources