k\in\{1,\dots,n\}.

(a+b)^{n+1}=(a+b)(a+b)^n=(a+b)\sum_{k=0}^n{n\choose k}a^kb^{n-k}=\sum_{k=0}^n{n\choose k}a^{k+1}b^{n+1-(k+1)}+\sum_{k=0}^n{n\choose k}a^kb^{n+1-k}