4 - Chapter 7 Use the following to answer questions 1-16 In...

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Chapter 7 Use the following to answer questions 1-16: In the questions below, describe each sequence recursively. Include initial conditions and assume that the sequences begin with a 1 . 1. a n = 5 n . Ans: a n = 5 a n 1 , a 1 = 5. 2. The Fibonacci numbers. Ans: a n = a n 1 + a n 2 , a 1 = a 2 = 1. 3. 0 , 1 , 0 , 1 , 0 , 1 , . Ans: a n = a n 2 , a 1 = 0, a 2 = 1. 4. a n = 1 + 2 + 3 + ... + n . Ans: a n = a n 1 + n , a 1 = 1. 5. 3 , 2 , 1 , 0 ,− 1 ,− 2 , . Ans: a n = a n 1 1, a 1 = 3. 6. a n = n ! . Ans: a n = na n 1 , a 1 = 1. 7. 1 / 2 , 1 / 3 , 1 / 4 , 1 / 5 , . Ans: 1 1 1 n n n a a + a = , a 1 = 1 / 2. 8. 0 . 1, 0 . 11, 0 . 111, 0 . 1111 , . Ans: a n = a n 1 + 1 / 10 n , a 1 = 0 . 1. 9. 1 2 , 2 2 , 3 3 , 4 2 , . Ans: a n = a n 1 + 2 n 1, a 1 = 1. 10. 1 , 111 , 11111 , 1111111 , . Ans: a n = 100 a n 1 + 11. 11. a n = the number of subsets of a set of size n . Ans: a n = 2 a n 1 , a 1 = 2. 12. 1 , 101 , 10101 , 1010101 , . Ans: a n = 100 a n 1 + 1, a 1 = 1. Page 88
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13. a n = the number of bit strings of length n with an even number of 0s. Ans: a n = a n 1 + 2 n 2 , a 1 = 1. 14. a n = the number of bit strings of length n that begin with 1. Ans: a n = 2 a n 1 , a 1 = 1. 15. a n = the number of bit strings of length n that contain a pair of consecutive 0s. Ans: a n = a n 1 + a n 2 + 2 n 2 , a 1 = 0, a 2 = 1. 16. a n = the number of ways to go down an n -step staircase if you go down 1, 2, or 3 steps at a time. Ans: a n = a n 1 + a n 2 + a n 3 , a 1 = 0, a 2 = 1, a 3 = 1. 17. Verify that a n = 6 is a solution to the recurrence relation a n = 4 a n 1 3 a n 2 . Ans: 4 6 3 6 = 1 6 = 6. 18. Verify that a n = 3 n is a solution to the recurrence relation a n = 4 a n 1 3 a n 2 . Ans: 4 3 n 1 3 3 n 2 = 4 3 n 1 3 n 1 = 3 3 n 1 = 3 n . 19. Verify that a n = 3 n + 4 is a solution to the recurrence relation a n = 4 a n 1 3 a n 2 . Ans: 4 3 n + 3 3 3 n + 2 = 4 3 n + 3 3 n + 3 = 3 3 n + 3 = 3 n + 4 . 20. Verify that a n = 3 n + 1 is a solution to the recurrence relation a n = 4 a n 1 3 a n 2 . Ans: 4(3 n 1 + 1) 3(3 n 2 + 1) = 4 3 n 1 3 n 1 + 4 3 = 3 n 1 (4 1) + 1 = 3 n + 1. 21. Verify that a n = 7 3 n π is a solution to the recurrence relation a n = 4 a n 1 3 a n 2 . Ans: 4(7 3 n 1 ) 3(7 3 n 2 ) = 28 3 n 1 7 3 n 1 4 + 3 = 7 3 n . Use the following to answer questions 22-26: In the questions below find a recurrence relation with initial condition(s) satisfied by the sequence. Assume a 0 is the first term of the sequence. 22. a n = 2 n . Ans: a n = 2 a n 1 , a 0 = 1. 23. a n = 2 n + 1. Ans: a n = 2 a n 1 1, a 0 = 2. 24. a n = ( 1) n . Ans: a n = a n 1 , a 0 = 1. 25. a n = 3 n 1. Ans: a n = a n 1 + 3, a 0 = 1. Page 89
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26. 2 n = a . Ans: a n = a n 1 , 0 2 a = . 27. You take a job that pays $25,000 annually. (a) How much do you earn n years from now if you receive a three percent raise each year? (b) How much do you earn n years from now if you receive a five percent raise each year?
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4 - Chapter 7 Use the following to answer questions 1-16 In...

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