Chapter 6.2, Problem 70E

Throughout this exercise, consider the Fibonacci sequence $f_0,f_1,f_2,\mathrm{...}$ recursively defined by $f_0=0,f_1=1$ ,and $f_{n+2}=f_n+f_{n+1}$ for all $n=0,1,2,\mathrm{...}$
a. Find the Fibonacci numbers $f_0,f_1,\mathrm{....}{f}_8.$
b. Consider the matrix $A=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$ . Prove by induction(seeAppendixB.1)that $A^n=\left[\begin{array}{cc}{f}_{n+1}& f_n\\ f_n& f_{n-1}\end{array}\right]$ for all $n=1,2,\mathrm{....}$
c. Show that $f_{n+1}{f}_{n-1}-f_n^2={\left(-1\right)}^n$ . This equation is known as Cassini‘s identity; it was discovered by the Italian/French mathematician and astronomer Giovanni Domenico Cassini, 1625-1712.