Chapter 4.3, Problem 68E

Consider the linear space V of all infinite sequences of real numbers. We define the subset W of V consisting of all sequences $\left(x_0,x_1,x_2,\mathrm{...}\right)$ such that $x_{n+2}=x_{n+1}+6x_n$ for all $n\ge 0$ .
a. Show that W is a subspace of V.
b. Determine the dimension of W.
c. Does W contain any geometric sequences of the form $\left(1,c,c^2,c^3,\mathrm{...}\right)$ , for some constant c? Find all such sequences in W.
d. Can you find a basis of W consisting of geometric sequences?
e. Consider the sequence in W whose first two terms are $x_0=0,x_1=1$.Find $x_2,x_3,x_4$.Find a closed formula for the nth term $x_n$ of this sequence. Hint: Write this sequence as a linear combination of the sequences you found in part (d).