Tree 1: A treelike figure is drawn in the plane, as shown in Figure 14-4a. The first

Chapter 14, Problem C1

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Tree 1: A treelike figure is drawn in the plane, as shown in Figure 14-4a. The first year, the tree grows a trunk 2 m long. The next year, two branches, each 1 m long, grow at right angles to each other from the top of the trunk, symmetrically to the line of the trunk. In subsequent years, each branch grows two new branches, each half as long as the preceding branch. Figure 14-4a a. Show that the lengths of the branches form a geometric sequence. b. Find the height of the tree after 2 years, 3 years, and 4 years. c. Show that the height of the tree is the partial sum of two geometric series. What is the common ratio of each? d. If the tree keeps growing like this forever, i. What limit will the height approach? ii. What limit will the width approach? iii. What limit will the length of each branch approach? iv. What limit will the total length of all branches approach? v. How close to the ground will the lowest branches come?