34. Sides of a Polygon The series of sketches below starts with an equilateral triangle having sides of length 1. In the following steps, equilateral triangles are constructed on each side of the preceding figure. The length of the sides of each new triangle is 1 3 the length of the sides of the preceding triangles. Develop a formula for the number of sides of the nth figure. Use mathematical induction to prove your answer. 35. Perimeter Find the perimeter of the nth figure in Exercise 34. 36. Area Show that the area of the nth figure in Exercise 34 is 23 c 2 5 - 3 20 a 4 9b n-1d . 37. Tower of Hanoi A pile of n rings, each ring smaller than the one below it, is on a peg. Two other pegs are attached to a board with this peg. In the game called the Tower of Hanoi puzzle, all the rings must be moved to a different peg, with only one ring moved at a time, and with no ring ever placed on top of a smaller ring. Find the least number of moves (in terms of n) that would be required. 38. Tower of Hanoi Prove the result of Exercise 37 using mathematical induction. Write the first five terms of each sequence. State whether the sequence is arithmetic, geometric, or neither. 1. an = -4n + 2 2. an = -2 a- 1 2b n 3. a1 = 5, a2 = 3, an = an-1 + 3an-2, for n Ú 3 Solve each problem. 4. An arithmetic sequence has a1 = -6 and a9 = 18. Find a7. 5. Find the sum of the first ten terms of each series described. (a) arithmetic, a1 = -20, d = 14 (b) geometric, a1 = -20, r = - 1 2 6. Evaluate each sum that converges. Identify any that diverge. (a) a30 i=11-3i + 62 (b) a ∞ i=1 2i (c) a ∞ i=1 a 3 4b i 7. Write the binomial expansion of 1x - 3y25. 8. Find the fifth term of the binomial expansion of A4x - 1 2 yB 5 . 9. Evaluate each expression. (a) 9! (b) a 10 4 b 10. Let Sn represent the following statement, and use mathematical induction to prove that Sn is true for every positive integer n. 6 + 12 + 18 + g+ 6n = 3n1n + 12 Chapter 11 Quiz (Sections 11.1–11.5) 1087 CHAPTER 11 Quiz
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