Chapter 5.6, Problem 25ES

In 24-34, $F_a,F_1,F_2,\mathrm{...}$is the Fibonacci sequence.

The Fibonacci sequence satisfies the recurrence relation $F_k=F_{k-1}+F_{k-2}$ , For every integer $k\ge 2.$

a. Explain why the following is true: $F_{k+1}=F_k+F_{k-1}$ for each integer $k\ge 1.$

b. Write an equation expressing $F_{k+2}$ in terms of $F_{k+1}$ and $F_k$ .

c. Write an equation expressing $F_{k+3}$ in terms $F_{k+2}$ and $F_{k+1.}$