An ant is standing on a number line at point A. It walks halfway to point B and turns ar...

An ant is standing on a number line at point A. It walks halfway to point B and turns around. Then it walks halfway back to point A, turns around again, and walks halfway to point B. It continues to do this indefinitely. Let point A be at 0 and point B be at 1. The ant’s walk is made up of a sequence of overlapping line segments. Let x₁ record the positions of the left-hand endpoints of these segments and x₂ their right-hand endpoints. (Thus, we begin with and so on.) Figure 2.33 shows the start of the ant’s walk.

(a) Make a table with the first six values of [x₁, x₂] and plot the corresponding points on x₁, x₂ coordinate axes.

(b) Find two linear equations of the form that determine the new values of the endpoints at each iteration. Draw the corresponding lines on your coordinate axes and show that this diagram would result from applying the Gauss-Seidel method to the system of linear equations you have found. (Your diagram should resemble Figure 2.27 on page 126.)

(c) Switching to decimal representation, continue applying the Gauss-Seidel method to approximate the values to which x₁ and x₂ are converging to within 0.001 accuracy.

(d) Solve the system of equations exactly and compare your answers. Interpret your results.