Chapter 9 Recurrence Relations and Generating Functions 492 Therefore, at the start (when Douglas has a dollars), the probability of Doug winning the game is а a +b' and the probability of Jennifer's winning the game is а 1- a +b 11 a +b Hence a player's probability of winning is proportional to the amount of mo with which he or she starts the game. EXERCISES 9.3 In Exercises 1-24, find an explicit formula for sn if so, S1, S2, .. . is a sequence satisfying the given recurr relation and initial conditions. 5sn-1-4, so= 1 4. Sn= 1.5sn-1 - 1, so 4 6. Sn= Sn-1- 10, so 32 8. Sn=-2sn-1, So = -5 2. Sn= I. Sn Sn-1+3, so 2 = 4s-1, S0 5 5. Sn 3. Sn= -Sn-1 +6, s0 =-4 7. Sn 3sn-1-8, so = 3 9. Sn= Sn-1-5, so = 100 11. Sn =-2s-1- 9, so 7 10. Sn =-Sn-1 + 7, so = 1 12. sn 10s-1 - 45, so = 2 13. Sn Sn-1 + 2s-2, S0 = 9, s1 = 0 14. Sn-2sn-1 - Sn-2, So= 3, s1 = 1 16. Sn 4s-2, so = -1, S1 = --14 18. sn 6s-1-9s,-2, So= 1, S1 = 9 20. Sn =-8sn-1- 15sp-2, So 2, s1 2 22. Sn 10s-1 -24s,-2, So = 1, S1 = 0 24. Sn 4sn-1- 4sn-2, So =-3, s1 = 4 15. Sn 8s-1- 16s-2, So = 6, S1 = 20 17. Sn9sn-2, So = 1, s1 = 9 = -4sn-1-4sn-2, So = -4, s1 = 2 10sm-1 25s-2, so = -7 , s1= -15 23. Sn-5sn-1 - 4s-2, S0 3, s1 = 15 sOUL 19. Sn 21. Sn= 25. In order to combat hypertension, Mr. Lorenzo is to take a capsule containing 25 mg of a drug each mor after awaking. During the next 24 hours, 20 percent of the amount of the drug in the body is eliminated. (a) Write a difference equation and initial conditions giving the amount of the drug in Mr. Lorenzo's immediately after he takes the nth capsule. (b) How much of the drug will there be in Mr. Lorenzo's body immediately after he takes the eighth cape (c) To what level will this drug eventually accumulate in his body? 26. In Mayville, 90 percent of the existing dog licenses are reissued each year, and 1200 new licenses are iss In 2000 there were 15,000 dog licenses issued. (a) Write a difference equation and initial conditions describing the number of dog licenses Mayville issue n years after 2000.
Chapter 9 Recurrence Relations and Generating Functions 492 Therefore, at the start (when Douglas has a dollars), the probability of Doug winning the game is а a +b' and the probability of Jennifer's winning the game is а 1- a +b 11 a +b Hence a player's probability of winning is proportional to the amount of mo with which he or she starts the game. EXERCISES 9.3 In Exercises 1-24, find an explicit formula for sn if so, S1, S2, .. . is a sequence satisfying the given recurr relation and initial conditions. 5sn-1-4, so= 1 4. Sn= 1.5sn-1 - 1, so 4 6. Sn= Sn-1- 10, so 32 8. Sn=-2sn-1, So = -5 2. Sn= I. Sn Sn-1+3, so 2 = 4s-1, S0 5 5. Sn 3. Sn= -Sn-1 +6, s0 =-4 7. Sn 3sn-1-8, so = 3 9. Sn= Sn-1-5, so = 100 11. Sn =-2s-1- 9, so 7 10. Sn =-Sn-1 + 7, so = 1 12. sn 10s-1 - 45, so = 2 13. Sn Sn-1 + 2s-2, S0 = 9, s1 = 0 14. Sn-2sn-1 - Sn-2, So= 3, s1 = 1 16. Sn 4s-2, so = -1, S1 = --14 18. sn 6s-1-9s,-2, So= 1, S1 = 9 20. Sn =-8sn-1- 15sp-2, So 2, s1 2 22. Sn 10s-1 -24s,-2, So = 1, S1 = 0 24. Sn 4sn-1- 4sn-2, So =-3, s1 = 4 15. Sn 8s-1- 16s-2, So = 6, S1 = 20 17. Sn9sn-2, So = 1, s1 = 9 = -4sn-1-4sn-2, So = -4, s1 = 2 10sm-1 25s-2, so = -7 , s1= -15 23. Sn-5sn-1 - 4s-2, S0 3, s1 = 15 sOUL 19. Sn 21. Sn= 25. In order to combat hypertension, Mr. Lorenzo is to take a capsule containing 25 mg of a drug each mor after awaking. During the next 24 hours, 20 percent of the amount of the drug in the body is eliminated. (a) Write a difference equation and initial conditions giving the amount of the drug in Mr. Lorenzo's immediately after he takes the nth capsule. (b) How much of the drug will there be in Mr. Lorenzo's body immediately after he takes the eighth cape (c) To what level will this drug eventually accumulate in his body? 26. In Mayville, 90 percent of the existing dog licenses are reissued each year, and 1200 new licenses are iss In 2000 there were 15,000 dog licenses issued. (a) Write a difference equation and initial conditions describing the number of dog licenses Mayville issue n years after 2000.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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Transcribed Image Text:Chapter 9 Recurrence Relations and Generating Functions
492
Therefore, at the start (when Douglas has a dollars), the probability of Doug
winning the game is
а
a +b'
and the probability of Jennifer's winning the game is
а
1-
a +b
11
a +b
Hence a player's probability of winning is proportional to the amount of mo
with which he or she starts the game.
EXERCISES 9.3
In Exercises 1-24, find an explicit formula for sn if so, S1, S2, .. . is a sequence satisfying the given recurr
relation and initial conditions.
5sn-1-4, so= 1
4. Sn= 1.5sn-1 - 1, so 4
6. Sn= Sn-1- 10, so 32
8. Sn=-2sn-1, So = -5
2. Sn=
I. Sn Sn-1+3, so 2
= 4s-1, S0 5
5. Sn
3. Sn=
-Sn-1 +6, s0 =-4
7. Sn 3sn-1-8, so = 3
9. Sn= Sn-1-5, so = 100
11. Sn =-2s-1- 9, so 7
10. Sn =-Sn-1 + 7, so = 1
12. sn
10s-1 - 45, so = 2
13. Sn
Sn-1 + 2s-2, S0 = 9, s1 = 0
14. Sn-2sn-1 - Sn-2, So= 3, s1 = 1
16. Sn 4s-2, so = -1, S1 = --14
18. sn 6s-1-9s,-2, So= 1, S1 = 9
20. Sn =-8sn-1- 15sp-2, So 2, s1 2
22. Sn 10s-1 -24s,-2, So = 1, S1 = 0
24. Sn 4sn-1- 4sn-2, So =-3, s1 = 4
15. Sn 8s-1- 16s-2, So = 6, S1 = 20
17. Sn9sn-2, So = 1, s1 = 9
= -4sn-1-4sn-2, So = -4, s1 = 2
10sm-1 25s-2, so = -7 , s1= -15
23. Sn-5sn-1 - 4s-2, S0 3, s1 = 15
sOUL
19. Sn
21. Sn=
25. In order to combat hypertension, Mr. Lorenzo is to take a capsule containing 25 mg of a drug each mor
after awaking. During the next 24 hours, 20 percent of the amount of the drug in the body is eliminated.
(a) Write a difference equation and initial conditions giving the amount of the drug in Mr. Lorenzo's
immediately after he takes the nth capsule.
(b) How much of the drug will there be in Mr. Lorenzo's body immediately after he takes the eighth cape
(c) To what level will this drug eventually accumulate in his body?
26. In Mayville, 90 percent of the existing dog licenses are reissued each year, and 1200 new licenses are iss
In 2000 there were 15,000 dog licenses issued.
(a) Write a difference equation and initial conditions describing the number of dog licenses Mayville
issue n years after 2000.
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