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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 16

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online