Exercises 1 and 2 demonstrate that sometimes, if we are lucky, the form of an iterative problem may allow us to use a little insight to obtain an exact solution.
1. A narrow strip of paper 1 unit long is placed along a number line so that its ends are at 0 and 1. The paper is folded in half, right end over left, so that its ends are now at 0 and 1/2. Next, it is folded in half again, this time left end over right, so that its ends are at 1/4 and 1/2. Figure 2.32 shows this process. We continue folding the paper in half, alternating right-over-left and leftover- right. If we could continue indefinitely, it is clear that the ends of the paper would converge to a point. It is this point that we want to find.
(a) Let x1 correspond to the left-hand end of the paper and x2 to the right-hand end. 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 x2 = ax1 + band x1 = cx2 + d 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 point to which the ends of the paper are converging to within 0.00 1 accuracy.
(d) Solve the system of equations exactly and compare your answers.
2. 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 x1 = 0 and x2 = 1/2. Then we have x1 = 1/4 and x2 = 1/2, 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 x2 = ax1 + b and x1 = cx2 + d that determine the new values of the endpoints at each iteration. Draw the corresponding


Figure 2.32
Folding a strip of paper


Figure 2.33
The ant's walk
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.