Please help me with this word problem involving Fibonacci's sequence and power series.

5. In this exercise we will begin with a strange power series and then find its sum. The Fibonacci sequence {fn } is a famous sequence whose first few terms are fo = 0, f1 = 1, f2 = 1, f3 = 2, f4 = 3, f5 = 5, f6 = 8, f7 = 13, . .., where each term in the sequence after the first two is the sum of the preceding two terms. That is, fo = 0, f1 = 1 and for n 2 2 we have fn = fn-1+ fn-2. Now consider the power series F(x) = > frack k=0 We will determine the sum of this power series in this exercise. a. Explain why each of the following is true. i. XF(x) = > 2k-1 fk-1ack ii. x2 F(a) = ER-2 fk-220 k b. Show that F(x) - XF(x) - 22F(x) = 2.c. Now use the equation F(z) zF(z) zF(z) =z to find a simple form for F(z) that doesn't involve a sum. d. Use a computer algebra system or some other method to calculate the first 8 derivatives of * evaluated at 0. Why shouldn't the results 1z22 surprise you