In Exercises 1 through 6, give the first four terms and identify the given recurrence relation...

   

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In Exercises 1 through 6, give the first four terms and identify the given recurrence relation as linear homogeneous or not. If the relation is a linear homogeneous relation, give its degree. 1. a = 2.5a1, a = 4 2. b = -3b-12b-2, b =-2, b = 4 3. C=2" C-1, c = 3 4. d = nd-1, d = 2 5. e=5e-1+ 3, e = 1 6. 8n=8-1+8-2, 81 = 1, 82 = 3 7. Let A = {0, 1}. Give a recurrence relation for the number of strings of length n in A* that do not contain 01. 8. Let A={0, 1}. Give a recurrence relation for the number of strings of length n in A" that do not contain 111. 9. On the first of each month Mr. Martinez deposits $100 in a savings account that pays 6% compounded monthly. As- suming that no withdrawals are made, give a recurrence relation for the total amount of money in the account at the end of n months. In Exercises 12 through 17, use the technique of backtrack- ing to find an explicit formula for the sequence defined by the recurrence relation and initial condition(s). 12. a 2.5a-1, a = 4 = 13. b=5b-1+3, b = 3 14. C=C-1+n, c = 4 15. d=-1.1d-1, d = 5 16. en-en-1-2, e = 0 == 17. gn=ngn-1.81 = 6 In Exercises 18 through 23, solve each of the recurrence rela- tions. 18. an = 4a-1+5am-2, a = 2, a = 6 19. bm=-3b-1-2b-2, b = -2, b = 4 20. C=-6-1-9cm-2, c = 2.5, c = 4.7 21. d=4d-1-4d-2, d = 1, d = 7 22. e=2e-2, e = 2, e = 6 23. gn=2gn-1-2gm-2, g1 = 1, 82 = 4 24. Develop a general explicit formula for a nonhomogeneous recurrence relation of the form a = ra-+s, where r and s are constants. In Exercises 1 through 6, give the first four terms and identify the given recurrence relation as linear homogeneous or not. If the relation is a linear homogeneous relation, give its degree. 1. a = 2.5a1, a = 4 2. b = -3b-12b-2, b =-2, b = 4 3. C=2" C-1, c = 3 4. d = nd-1, d = 2 5. e=5e-1+ 3, e = 1 6. 8n=8-1+8-2, 81 = 1, 82 = 3 7. Let A = {0, 1}. Give a recurrence relation for the number of strings of length n in A* that do not contain 01. 8. Let A={0, 1}. Give a recurrence relation for the number of strings of length n in A" that do not contain 111. 9. On the first of each month Mr. Martinez deposits $100 in a savings account that pays 6% compounded monthly. As- suming that no withdrawals are made, give a recurrence relation for the total amount of money in the account at the end of n months. In Exercises 12 through 17, use the technique of backtrack- ing to find an explicit formula for the sequence defined by the recurrence relation and initial condition(s). 12. a 2.5a-1, a = 4 = 13. b=5b-1+3, b = 3 14. C=C-1+n, c = 4 15. d=-1.1d-1, d = 5 16. en-en-1-2, e = 0 == 17. gn=ngn-1.81 = 6 In Exercises 18 through 23, solve each of the recurrence rela- tions. 18. an = 4a-1+5am-2, a = 2, a = 6 19. bm=-3b-1-2b-2, b = -2, b = 4 20. C=-6-1-9cm-2, c = 2.5, c = 4.7 21. d=4d-1-4d-2, d = 1, d = 7 22. e=2e-2, e = 2, e = 6 23. gn=2gn-1-2gm-2, g1 = 1, 82 = 4 24. Develop a general explicit formula for a nonhomogeneous recurrence relation of the form a = ra-+s, where r and s are constants.