Chapter 8 Answers 1 Chapter 8 Answers Lesson 8.1 1 1 . . a. This a bipartite graph. b. 1, 4, 9 c. Number of Couples Number of Handshakes Recurrence Relation 1 1 — 2 4 H 2 = H 1 + 3 3 9 H 3 = H 2 + 5 4 16 H 4 = H 3 + 7 5 25 H 5 = H 4 + 9 d. 2n – 1 e. H n = H n–1 + 2n – 1 2 2 . . a. Couples Handshakes 1 0 2 2 3 6 4 12 b. 2n – 2. c. H n = H n–1 + 2n – 2 3 3 . . a. i. H n = H n–1 + 3 ii. H n = (H n–1 ) × 2 iii. H n = H n–1 + n iv. H n = (H n–1 ) × n b. 16, 64, 36, 5040
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b. Vn = (Vn–1) × 1.05 c. The differences never become constant.
8. a.
Year Deer 2010 50 2011 52 2012 54.08 2013 56.24
b. In 2058 c. Tn = 1.04Tn–1 d. The differences never become constant.
9. A recurrence relation is Sn = Sn–1 + n. A closed-form formula is Sn = n2 + n
2.
10. For a third-degree polynomial, the third differences are constant. Each third difference is 6 times the leading coefficient.
Chapter 8 Answers 7
Lesson 8.3 11.. a. i. Arithmetic
ii. Geometric iii. Geometric iv. Neither v. Geometric vi. Neither
b. i. Hn = Hn–1 + 3
ii. Hn =
Hn!1
2
iii. Hn = (Hn–1) × 1.2 iv. Hn = Hn–1 + Hn–2
v. Hn = (Hn–1) × 0.1 vi. Hn = Hn–1 + n – 1
c. i. Hn = 2 + 3(n – 1) ii. Hn = 64(0.5n–1) iii. Hn = 10(1.2n–1) v. Hn = 0.3(0.1n–1)
2. 0 3. There is no fixed point. 4. a. Cn = Cn–1 + 0.95
b. Cn = 2 + 0.95(n – 1) c. $76,500, $61,500
5. a. Mn = (Mn–1) × 1.048 b. Mn = 5000(1.048n) c. $5,755.11 d. 0.4%
e. Mn = (Mn–1) × 1.004 f. Mn = 5000(1.004n) g. $5,772.76 h. Yearly = $5,755.11. Monthly = $5,772.76 Monthly compounding produces $17.65 more interest than yearly
The amounts are generated with the mixed recurrence relation An = 1.07(An–1) + 1000. The final balance is approximately what is claimed in the article.
Chapter 8 Answers 16
b. The following table tracks the account for 20 years
The first five years are generated with the mixed recurrence relation An = 1.09(An–1) + 2,000; the remaining years with the geometric recurrence relation An = 1.09(An–1).
The amounts are generated with the mixed recurrence relation An = 1.09(An–1) + 2,000.
Sasha is wrong. Plan A has over three times the amount that Plan B has. 9. If a = 1, the recurrence relation is arithmetic. An arithmetic sequence does not
have a fixed point (unless the constant difference is 0). The closed form for the recurrence relation can be found by applying previous results for arithmetic sequences.
Chapter 8 Answers 21
Lesson 8.6 1.
2. a. 4, repelling
b. 2.5, neither c. 6, attracting d. 12, attracting
3. Attracting when |a| < 1, repelling when |a| > 1
Chapter 8 Answers 22
4. a. 1 b. 3 c.
n tn 1 1 2 5 3 13
d. tn = 2tn–1 + 3 e. The fixed point is –3. It can be found algebraically by solving the equation
x = 2x + 3. 5. a. The population can be modeled by the recurrence relation Pn = 1.04Pn–1 + b,
where b is the annual harvest. The fixed point is
b1!1.04
= –25b. Thus, the
closed form is (12000 + 25b)1.04n–1 – 25b. Setting this equal to 10,000, assigning n the value 10, and solving for b gives approximately –669. Thus issuing 670 permits annually for 10 years would bring the population to about 10,000 in 10 years, assuming the each permit is fulfilled.
b. Approximately 400 permits a year would hold the population constant. c. The plans seems reasonable assuming the nearly all permits are fulfilled.
However, the decrease from 670 to 400 permits at the end of 10 years could prove unpopular with hunters.
6. a.
n tn 1 0 2 4 3 –4 4 –4 5 –4 6 –4
Chapter 8 Answers 23
b.
c. The behavior is unpredictable. d. 2 is a fixed point. e. The terms diverge. f. 2, –4 g. Attracted when t1 = 0 or 2; repelled when t1 = 5; unpredictable when t1 = 1
7. a. tn = 1+ 0.1 1! tn!1
10"#$
%&'
"#$
%&'
tn!1
b. Approaches 10,000 but never reaches it c. The population approaches 10,000. d. 0 and 10,000 are the fixed points.
Chapter 8 Answers 24
Chapter 8 Review 1. A reasonable summary should include the following points:
A description of recurrence relations, perhaps with a few examples, including recursive and closed-form representations.
A discussion of finite difference techniques for finding polynomial closed forms.
Arithmetic and geometric recurrence relations, including closed forms. Mixed recurrence relations, including fixed points and their role in closed
forms. Cobweb diagrams, repelling and attracting fixed points. Important applications, particularly financial applications.
15. a. Bn = 1.008(Bn–1) – 230 b. Bn = –17,750(1.008)n + 28,750 c. 61 months d. $3,030 e. $352.88
16. a. Rn = 0.9Rn–1 b. Rn = 1000(0.9n) c. About 6.58 minutes or 6 minutes, 35 seconds
17. a. Vn = 1.5Vn–1 + 4,000 b. Vn = 16,000(1.5)n–1 – 8000
Chapter 8 Answers 27
18.
19. a. An = 0.4An–1 + 500
b. The amount of medication in the body stabilizes at 833 mg.
20
20
10
10 30
Chapter 8 Answers 28
c. The cobweb would be attracted to the point (833.33, 833.33), which is the intersection of y = x and y = 0.4x + 500, as shown in this figure.
d. The amount in the body reaches the stable value (833 mg, in this case) more
quickly. The stable value is probably near the optimal dosage of the drug. 20. a. Cn = 0.8Cn–1 + 1
b. The daily concentration sequence is 2, 2.6, 3.08, 3.464, … .The concentration quickly exceeds the recommended maximum of 3 ppm and gradually approaches 5 ppm.
c. Sample answer: a daily addition of 0.4 ppm. The recurrence relation Cn = 0.8Cn–1 + 0.4 has a fixed point of 2. Daily additions of 0.4 ppm keep the concentration at a constant 2 ppm, which is the middle of the recommended range.