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Ch 9 Solutions - 3 2 3 3 3 4 3 … 11 The Golden Ratio is a...

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Chapter 9 Practice Problems – Solutions 1. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 The sequence progresses by adding the two previous terms in the sequence to obtain the next number. 2. 1, 3, 4, 7, 11, 18, 29, 47, 76, 123 The Lucas Numbers 3. a. 39,088,169 b. 63,245,986 4. a. 2,178,309 b. 5,702,887 5. 1 + 2 + 5 + 13 + 34 + 89 = 144 6. Leonard Euler discovered the explicit formula for finding Fibonacci numbers in 1736. However, Jacques Binet rediscovered the explicit formula and the formula, known as Binet’s formula, is named after him. 7. A recursive rule uses the numbers in the sequence to obtain additional numbers. An explicit rule uses a formula to calculate numbers in the sequence. You do not need to know any of the previous numbers in the sequence to find a number using an explicit rule. Just substitute! 8. a. 1, 4, 4, 16, 64, 1024, 65,536 b. 2, 5, 9, 19, 37, 75, 149 9. a. 66 b. 224 10. a. 25, 29, 33 – add 4 to the previous number b. 216, 343, 512 – perfect cubes (1
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Unformatted text preview: 3 , 2 3 , 3 3 , 4 3 , …) 11. The Golden Ratio is a solution to the equation x x 2 1 = + . The number has been traced back to ancient Greek philosophers. It is represented by the Greek letter φ (phi). It’s exact representation is 1 5 2 + . It’s approximate value is 1.618. 12. The Golden Ratio is related to the Fibonacci numbers because it is seen in Binet’s formula for finding the Fibonacci numbers. Also, as the Fibonacci numbers get larger, the ratio of consecutive Fibonacci numbers appears to approach the Golden Ratio. 13. The Divine Proportion, The Golden Number, The Golden Section 14. A golden rectangle is a rectangle whose sides are in the proportion of the golden ratio. 15. A Fibonacci rectangle is a rectangle whose sides are consecutive Fibonacci numbers. Examples (not drawn to scale): 16. Some examples of gnomonic growth include tree rings and a chambered nautilus. 2 1 13 21 55 89...
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