Rising Sum of Binomial Coefficients

Theorem
Let $n \in \Z$ be an integer such that $n \ge 0$.

Then:
 * $\displaystyle \sum_{j=0}^m \binom {n + j} n = \binom {n+m+1} {n+1} = \binom {n+m+1} m$

where $\displaystyle \binom n k$ denotes a binomial coefficient.

That is:
 * $\displaystyle \binom n n + \binom {n+1} n + \binom {n+2} n + \cdots + \binom {n+m} n = \binom {n+m+1} {n+1} = \binom {n+m+1} m$

Corollary

 * $\displaystyle \sum_{j=0}^m \binom {n + j} j = \binom {n+m+1} m$

That is:
 * $\displaystyle \binom n 0 + \binom {n+1} 1 + \binom {n+2} 2 + \cdots + \binom {n+m} m = \binom {n+m+1} m$

Proof
Proof by induction:

Let $n \in \Z$.

For all $m \in \N$, let $P \left({m}\right)$ be the proposition:
 * $\displaystyle \sum_{j=0}^m \binom {n + j} n = \binom {n+m+1} {n+1}$

$P(0)$ is true, as this just says $\binom n n = \binom {n+1} {n+1}$.

But $\displaystyle \binom n n = \binom {n+1} {n+1} = 1$ from the definition of a binomial coefficient.

Basis for the Induction
$P(1)$ is the case:

So:
 * $\displaystyle \sum_{j=0}^1 \binom {n + j} n = \binom {n + 2} {n + 1}$ and $P(1)$ is seen to hold.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k+1}\right)$ is true.

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

Then we need to show:
 * $\displaystyle \sum_{j=0}^{k+1} \binom {n + j} n = \binom {n+k+2} {n+1}$.

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \sum_{j=0}^m \binom {n + j} n = \binom {n+m+1} {n+1}$

Finally, we note that $\displaystyle \binom {n+m+1} {n+1} = \binom {n+m+1} m$ from Symmetry Rule for Binomial Coefficients.

Marginal cases
Just to make sure, it is worth checking the marginal cases:

n = 0
When $n = 0$ we have:

So the theorem holds for $n = 0$.

n = 1
When $n = 1$ we have:

So the theorem holds for $n = 1$.

Proof of Corollary
From the Symmetry Rule for Binomial Coefficients we have:
 * $\displaystyle \binom {n + j} n = \binom {n + j} j$

and
 * $\displaystyle \binom {n+m+1} {n+1} = \binom {n+m+1} m$

Hence the result.