282 CHAPTER 5 Number Theory and the Real Number System Consider the following numbers found in nature: the number of petals on a daisy, the number of sunflower seeds in a sunflower, the number of scales on a pineapple, the number of ancestors of a male bee. In this section, we will determine that all these numbers are part of a special sequence of numbers called the Fibonacci sequence. The Fibonacci Sequence and the Golden Ratio SECTION 5.8 LEARNING GOALS Upon completion of this section, you will be able to: 7 Identify the Fibonacci sequence. 7 Understand the relationship between Fibonacci numbers and the golden ratio. Why This Is Important The Fibonacci sequence and the related topic of the golden ratio appear in many places in nature and play an important role in geometry, art, and music. The Fibonacci Sequence Our discussion of sequences would not be complete without mentioning a sequence known as the Fibonacci sequence. The sequence is named after Leonardo of Pisa, also known as Fibonacci. He was one of the most distinguished mathematicians of the Middle Ages. This sequence is first mentioned in his book Liber Abacci (Book of the Abacus), which contained many interesting problems, such as: “A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is assumed that every month each pair begets a new pair which from the second month becomes productive?” The solution to this problem (Fig. 5.10) led to the development of the sequence that bears its author’s name: the Fibonacci sequence. The sequence is shown in Table 5.1. The numbers in the columns titled Pairs of Adults form the Fibonacci sequence. Recreational Mathematics 74. Martingale System See the Recreational Math box on page 277. Use the Martingale system to answer the following questions. a) Assume a player is using a $10 standard bet and loses five times in a row. How much money should the player bet in the sixth round? How much money has the player lost at the end of the fifth round? $320, $310 b) Assume a player is using a $10 standard bet and loses 10 times in a row. How much money should the player bet in the eleventh round? How much money has the player lost at the end of the tenth round? $10,240; $10,230 c) Why is this a risky strategy? * Research Activity 75. Series A topic generally associated with sequences is series. a) Research series and explain what a series is and how it differs from a sequence. Also write a formal definition of series. Give examples of different kinds of series. b) Write the arithmetic series associated with the arithmetic sequence 1, 4, 7, 10, 13, .… c) Write the geometric series associated with the geometric sequence 3, 6, 12, 24, 48, .… d) What is an infinite geometric series? e) Determine the sum of the terms of the infinite geometric series + + + + + 1 1 2 1 4 1 8 1 16 . *See Instructor Answer Appendix Johan Larson/Shutterstock
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