Chapter 4, Problem 4P

(a)

Program Plan Intro

To show that $F(z)=z+zF(z)+z^{2}F(z)$ .

(b)

Program Plan Intro

To show that $F(z)=\frac{z}{1-z-z^2}=\frac{1}{\sqrt{5}}(\frac{1}{1-\varphi z}-\frac{1}{1-{\varphi }^{\text{'}}z}).$

(c)

Program Plan Intro

To show that $F(z)={\displaystyle \sum _{}^{}\frac{1}{\sqrt{5}}({\varphi }^i-{\varphi }^{\text{'}i})z^i.}$

(d)

Program Plan Intro

To show that $Fi=\frac{{\varphi }^i}{\sqrt{5}}$ for $i>0$ by using $F(z)={\displaystyle \sum _{}^{}\frac{1}{\sqrt{5}}({\varphi }^i-{\varphi }^{\text{'}i})z{.}^i}$