Chapter 7.5, Problem 16Q

Solution

To state: The first three iterates of the function for the given initial value: $f\left(x\right)=5x-7,x_0=-3$

The resultant iterates are $-22,-117,\text{and}-592$ .

Given information:

The given function and initial value is: $f\left(x\right)=5x-7,x_0=-3$

Explanation:

Consider the given function,

  $f\left(x\right)=5x-7$

The given initial term of the function is: $x_0=-3$

The first iterate is:

  $\begin{array}{c}{x}_1=f\left(x_0\right)\\ =f\left(-3\right)\\ =5\left(-3\right)-7\\ =-15-7\\ =-22\end{array}$

Similarly, find the remaining 2 iterates,

  $\begin{array}{c}{x}_2=f\left(x_1\right)\\ =f\left(-22\right)\\ =5\left(-22\right)-7\\ =-110-7\\ =-117\\ x_3=f\left(x_2\right)\\ =f\left(-117\right)\\ =5\left(-117\right)-7\\ =-585-7\\ =-592\end{array}$

Therefore, the first three iterates are $-22,-117,\text{and}-592$ .