Consider the sequence ( f0, f1, f2, . . .) recursively defined by f0 = 0, f1 = 1, and fn...

Consider the sequence ( f₀, f₁, f₂, . . .) recursively defined by f₀ = 0, f₁ = 1, and f_n = f_n−2 + f_n−1 for all n = 2, 3, 4, . . . . This is known as the Fibonacci sequence; for some historical context, see Exercise 48 of Section 7.3.

In this exercise you are invited to derive a closed formula for f_n, expressing f_n in terms of n, rather than recursively in terms of f_n−1 and f_n−2. Another derivation of this closed formula will be presented in Exercise 7.3.48b.

a. Find the terms f₀, f₁, . . . , f₉, f₁₀ of the Fibonacci sequence.

b. In the space V of all infinite sequences of real numbers (see Example 5), consider the subset W of all sequences (x₀, x₁, x₂, . . .) that satisfy the recursive equation x_n = x_n−2 + x_n−1 for all n = 2, 3, 4, . . . . Note that the Fibonacci sequence belongs to W. Show that W is a subspace of V, and find a basis of W (write the first five terms x₀, . . . , x₄ of each sequence in your basis). Determine the dimension of W.

c. Find all geometric sequences of the form (1, r, r ², . . .) in W. Can you form a basis of W consisting of such sequences? (Be prepared to deal with irrational numbers.)

d. Write the Fibonacci sequence as a linear combination of geometric sequences. Use your answer to find a closed formula for