Chapter 8.4, Problem 4E

In Exercises 3-8, by a closed form we mean an algebraic expression not involving a summation over a range of values or the use of ellipses.

Find a closed form for the generating function for each of these sequences. Assume a general form for the terms of the sequence, using the most obvious choice of such a Sequence.)

a) $-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,\mathrm{...}$

b) 1,3,9,27, 81, 243, 729, …c) $0,0,\text{3},-\text{3},\text{ 3},-\text{3},\text{ 3},-\text{3},\mathrm{...}$

d) 1,2,1,1,1,1,1,1,1,... e) $\left(\begin{array}{c}7\\ 0\end{array}\right),2\left(\begin{array}{c}7\\ 1\end{array}\right),2^2\left(\begin{array}{c}7\\ 2\end{array}\right),{\mathrm{....2}}^7\left(\begin{array}{c}7\\ 7\end{array}\right),0,0,0,0,\mathrm{....}$

f) $-\text{3},\text{3},-\text{3},\text{3},-\text{3},\text{3},\mathrm{....}$

g) $0,\text{ 1},-\text{2},\text{4},-\text{8},\text{16},-\text{32},\text{64,}\text{.}$ ..

h) 1,0,1,0,1,0,1,0,...