Consider the linear space V of all infinite sequences of real numbers. We define the subset W of V consisting of all sequences ( jc o, x\ , jc2 , ...) such that jc + 2 = xn+i + 6xn for all n > 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 (1, c, c2, c3, ...), for some constant cl 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 jco = 0, jc i = 1. Find jc2 , JC3, JC4. Find a closed formula for the nth term jc of this sequence. Hint: Write this sequence as a linear combination of the sequences you found in part (d).

- -1- _ Jgk _ A hjje(.,oC_m'-tc