How to add to summations with different limits

Question

$$\sum_{k=0}^n {n \choose k} x^{n+1-k}y^k+\sum_{k=1}^{n+1} {n \choose k-1}x^{n+1-k}y^k$$

i would like to know how to add the two summations above together including a explanation of how the limits of the two sums will change.

Answer 1

In this case you can give the summations identical limits:

$$\sum_{k=0}^n\binom{n}kx^{n+1-k}y^k=\sum_{k=0}^{n+1}\binom{n}kx^{n+1-k}y^k\;,$$

since $\binom{n}{n+1}=0$ by definition, and

$$\sum_{k=1}^{n+1}\binom{n}{k-1}x^{n+1-k}y^k=\sum_{k=0}^{n+1}\binom{n}{k-1}x^{n+1-k}y^k\;,$$

since $\binom{n}{-1}=0$ by definition. Thus,

$$\begin{align*} \sum_{k=0}^n\binom{n}kx^{n+1-k}y^k+\sum_{k=1}^{n+1}\binom{n}{k-1}x^{n+1-k}y^k&=\sum_{k=0}^{n+1}\binom{n}kx^{n+1-k}y^k+\sum_{k=0}^{n+1}\binom{n}{k-1}x^{n+1-k}y^k\\ &=\sum_{k=0}^{n+1}\left(\binom{n}k+\binom{n}{k-1}\right)x^{n+1-k}y^k\\ &=\sum_{k=0}^{n-1}\binom{n+1}kx^{n+1-k}y^k\\ &=(x+y)^{n+1}\;. \end{align*}$$