5.8 The Fibonacci Sequence and the Golden Ratio 287 Exercises Warm Up Exercises In Exercises 1– 6, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. The sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21,… is known as the ________ sequence. Fibonacci 2. The irrational number + 5 1 2 is known as the ________ number. Golden 3. In medieval times, people referred to the golden proportion as the ________ proportion. Divine 4. A rectangle whose ratio of its length to its width is equal to the golden number is known as a golden ________. Rectangle In Exercises 5 and 6, use the following diagram and assume that the ratio AB AC is equal to the ratio AC CB A B C 5. Each ratio AB AC and AC CB is referred to as a golden ________. Ratio 6. The proportion AB AC AC CB = is called the golden ________. Proportion Practice the Skills/Problem Solving In Exercises 7–14, determine whether the sequence is a Fibonaccitype sequence (each term is the sum of the two preceding terms). If it is, determine the next two terms of the sequence. 7. 1, 3, 4, 7, 11, 18,… Yes; 29, 47 8. 2, 4, 6, 10, 16, 26,… Yes; 42, 68 9. 1, 2, 2, 4, 8, 32,… No 10. 2, 3, 6, 18, 108, 1944,… No 11. 1, 1, 0, 1, 1, 2, − … Yes; 3, 5 12. 0, , ,2,3,5, π π π π π … Yes; 8 , 13 π π 13. 5, 10, 15, 25, 40, 65,… Yes; 105, 170 14. … , , , ,1,2, 1 4 1 4 1 2 3 4 1 4 Yes; 3 1 4 , 5 1 4 15. Write out the first 15 terms of the Fibonacci sequence. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 16. The sum of any six consecutive Fibonacci numbers is always divisible by 4. Select any six consecutive Fibonacci numbers and show that for your selection this statement is true. Answers will vary. 17. The sum of any 10 consecutive Fibonacci numbers is always divisible by 11. Select any 10 consecutive Fibonacci numbers and show that for your selection this statement is true. Answers will vary. 18. The greatest common factor of any two consecutive Fibonacci numbers is 1. Show that this statement is true for the first 15 Fibonacci numbers. * 19. Twice any Fibonacci number minus the next Fibonacci number equals the second number preceding the original number. Select a number in the Fibonacci sequence and show that this pattern holds for the number selected. Answers will vary. 20. For any four consecutive Fibonacci numbers, the difference of the squares of the middle two numbers equals the product of the smallest and largest numbers. Select four consecutive Fibonacci numbers and show that this pattern holds for the numbers you selected. Answers will vary. 21. a) Select any two one-digit (nonzero) numbers and add them to obtain a third number. Continue adding the two previous terms to get a Fibonacci-type sequence. Answers will vary. b) Form ratios of successive terms to show how they will eventually approach the golden number. Answers will vary. 22. a) Select any three consecutive terms of a Fibonacci sequence. Subtract the product of the terms on each side of the middle term from the square of the middle term. What is the difference? Answers will vary. b) Repeat part (a) with three different consecutive terms of the sequence. Answers will vary. c) Make a conjecture about what will happen when you repeat this process for any three consecutive terms of a Fibonacci sequence. Answers will vary. However, you should get 1 or 1. − 23. Pascal’s Triangle One of the most famous number patterns involves Pascal’s triangle. The Fibonacci sequence can be determined by using Pascal’s triangle. Can you explain how that can be done? A hint is shown. 1 11 5 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The sums of the numbers along the diagonals (parallel to the one shown) are Fibonacci numbers. 24. Lucas Sequence a) A sequence related to the Fibonacci sequence is the Lucas sequence. The Lucas sequence is formed in a manner similar to the Fibonacci sequence. The first two numbers of the Lucas sequence are 1 and 3. Write the first eight terms of the Lucas sequence. 1, 3, 4, 7, 11, 18, 29, 47 b) Complete the next two lines of the following chart. + = + = + = + = + = 1 23 1 3 4 2 5 7 3 8 11 5 13 18 8 21 29, 13 34 47 + = + = c) What do you observe about the first column in the chart in part (b)? It is the Fibonacci sequence. SECTION 5.8 *See Instructor Answer Appendix
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