1. For the following function $f(n)$, find $\Delta^{k} f(n)$.
(1) $f(n)=2^{n}+(-1)^{n}$.
(2) $f(n)=\frac{1}{n}$.
(3) $f(n)=n \cdot 2^{n}$.
(4) $f(n)=\cos (n x+a)$.
1. (1) $2^{n}+2^{k}(-1)^{n+k}$.
(2) $\frac{(-1)^{k} \cdot k!}{n(n+1) \cdots(n+k)}$.
(3) $(n+2 k) \cdot 2^{n}$.
(4) $2^{k} \sin ^{k} \frac{x}{2} \cos \left[n x+\frac{k(\pi+x)}{2}+a\right]$.