Sum of r+k Choose k up to n

Theorem
Let $r \in \R$ be a real number.

Then:
 * $\displaystyle \forall n \in \Z: n \ge 0: \sum_{k \mathop = 0}^n \binom {r + k} k = \binom {r + n + 1} n$

where $\displaystyle \binom r k$ is a binomial coefficient.

Proof
Proof by induction:

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition
 * $\displaystyle \sum_{k \mathop = 0}^n \binom {r + k} k = \binom {r + n + 1} n$

Basis for the Induction
$\map P 0$ is true, as $\dbinom r 0 = 1 = \dbinom {r + 1} 0$ by definition.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P m$ is true, then it logically follows that $\map P {m + 1}$ is true.

So this is our induction hypothesis:
 * $\displaystyle \sum_{k \mathop = 0}^m \binom {r + k} k = \binom {r + m + 1} m$

Then we need to show:
 * $\displaystyle \sum_{k \mathop = 0}^{m + 1} \binom {r + k} k = \binom {r + m + 2} {m + 1}$

Induction Step
This is our induction step:

So $\map P m \implies \map P {m + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall n \in \Z_{\ge 0}: \sum_{k \mathop = 0}^n \binom {r + k} k = \binom {r + n + 1} n$

Also see

 * Rising Sum of Binomial Coefficients, where the result is proved for integer $r$