Chapter 12, Problem 1FCCE

Explain how a sequence of numbers is generated by a difference equation of the form $y_n=a{y}_{n-1}+b,y_0$ specified.

To determine

How a sequence of numbers is generated by a difference equation of the form $y_n=a{y}_{n-1}+b,y_0$ specified.

Answer to Problem 1FCCE

Solution:

Sequence of numbers can be obtained by substituting different values for $\text{'}n\text{'}$ after knowing the values of a, b and $y_0$.

Explanation of Solution

Given Information:

Difference equation of the form $y_n=a{y}_{n-1}+b,y_0$ specified.

Consider the difference equation $y_n=0.2y_{n-1}+4.8,y_0=1$.

Put different values for $n$.

For $n=1$,

$\begin{array}{l}{y}_1=0.2\times 1+4.8\\ y_1=5\end{array}$

For $n=2$,

$\begin{array}{l}{y}_2=0.2\times 5+4.8\\ y_2=5.8\end{array}$

For $n=3$,

$\begin{array}{l}{y}_3=0.2\times 5.8+4.8\\ y_3=5.96\end{array}$

For $n=4$,

$\begin{array}{l}{y}_4=0.2\times 5.96+4.8\\ y_4=5.992\end{array}$

In this way, a sequence of numbers can be generated from any difference equation when a, b and $y_0$ is specified.