Define the binomial coefficients $\left(\begin{array}{l}n \\ k\end{array}\right)$, for integers $n, k$ satisfying $n \geqslant k \geqslant 0$. Prove directly from your definition that if $n>k \geqslant 0$ then

$$\left(\begin{array}{l} n \\ k \end{array}\right)+\left(\begin{array}{c} n \\ k+1 \end{array}\right)=\left(\begin{array}{c} n+1 \\ k+1 \end{array}\right)$$

and that for every $m \geqslant 0$ and $n \geqslant 0$,

$$\sum_{k=0}^{m}\left(\begin{array}{c} n+k \\ k \end{array}\right)=\left(\begin{array}{c} n+m+1 \\ m \end{array}\right)$$