Chapter 4.1, Problem 60E

Consider the sequence $\left(f_0,f_1,f_2\right)$ recursively defined by $f_0=0,f_1=1$ , and $f_n=f_{n-2}+f_{n-1}$ for all $n=2,3,\mathrm{4....}$.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 $f_0,f_1,\mathrm{...},f_9,f_{10}$ 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 $\left(x_0,x_1,x_2,\mathrm{...}\right)$ that satisfy the recursive equation $x_n=x_{n-2}+x_{n-1}$ for all $n=2,3,4,\mathrm{....}$ 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 $x_0,\mathrm{...},x_4$ of each sequence in your basis). Determine the dimension of W.
c. Find all geometric sequences of the form $\left(1,r,r^2,\mathrm{...}\right)$ 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 $f_n{.}^6$ .

e. Explain why f,, is the integer closest to $\frac{1}{\sqrt{5}}{\left(\frac{1+\sqrt{5}}{2}\right)}^n$ , for all $n=0,1,2,\mathrm{....}$ Use technology to find $f_{50}{.}^7$ .

f. Find $\underset{n\to 0}{\lim }\frac{f_n+1}{f_n}$ .