10-5 Recursion and Iteration
Find the first five terms of each sequence described.
1. a 1 = 16, a n + 1 = a n + 4
SOLUTION: The first five terms of the sequence are 16, 20, 24, 28 and 32.
3. a 1 = 5, a n + 1 = 3a n + 2
SOLUTION: The first five terms of the sequence are 5, 17, 53, 161 and 485.
Write a recursive formula for each sequence.
5. 3, 8, 18, 38, 78, …
SOLUTION: 7. FINANCING
Benbyfinanced
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$1500 rowing machine to help him train for the college rowing team. He could only
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make a $100 payment each month, and his bill increased by 1% due to interest at the end of each month.
10-5 Recursion and Iteration
7. FINANCING Ben financed a $1500 rowing machine to help him train for the college rowing team. He could only
make a $100 payment each month, and his bill increased by 1% due to interest at the end of each month.
a. Write a recursive formula for the balance owed at the end of each month.
b. Find the balance owed after the first four months.
c. How much interest has accumulated after the first six months?
SOLUTION: a. Here a 1 = 1500.
The recursive formula for the balance owed at the end of each month is
.
b. Substitute n = 1 in
.
The balance owed after the first four months will be $1154.87.
c. Find the balance owed after the first six months.
Amount paid for the first six months is $600.
So, after six months $900 should be paid if there is no interest at all.
After the first six months, the interest has accumulated to $77.08.
Find the first three iterates of each function for the given initial value.
9. f (x) = –4x + 2, x0 = 5
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SOLUTION: Page 2
So, after six months $900 should be paid if there is no interest at all.
the firstand
six Iteration
months, the interest has accumulated to $77.08.
10-5After
Recursion
Find the first three iterates of each function for the given initial value.
9. f (x) = –4x + 2, x0 = 5
SOLUTION: The first three iterates are –18, 74 and –294.
11. f (x) = 8x – 4, x0 = –6
SOLUTION: The first three iterates are –52, –420 and –3364.
CCSS PERSEVERANCE Find the first five terms of each sequence described.
13. a 1 = –9, a n + 1 = 2a n + 8
SOLUTION: The Manual
first five
termsbyofCognero
the sequence
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are –9, –10, –12, –16 and –24.
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first three
iterates
are –52, –420 and –3364.
10-5The
Recursion
and
Iteration
CCSS PERSEVERANCE Find the first five terms of each sequence described.
13. a 1 = –9, a n + 1 = 2a n + 8
SOLUTION: The first five terms of the sequence are –9, –10, –12, –16 and –24.
15. a 1 = –4, a n + 1 = 2a n + n
SOLUTION: The first five terms of the sequence are –4, –7, –12, –21 and –38.
17. a 1 = –2, a n + 1 = 5a n + 2n
SOLUTION: The first five terms of the sequence are –2, –8, –36, –174 and –862.
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19. a 1 = 4, a 2 = 5, a n + 2 = 4a n – 2a n + 1
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10-5The
Recursion
Iteration
first fiveand
terms
of the sequence are –2, –8, –36, –174 and –862.
19. a 1 = 4, a 2 = 5, a n + 2 = 4a n – 2a n + 1
SOLUTION: The first five terms of the sequence are 4, 5, 6, 8 and 8.
21. a 1 = 3, a 2 = 2x, an = 4a n – 1 – 3a n – 2
SOLUTION: The first five terms of the sequence are 3, 2x, 8x – 9, 26x – 36 and 80x –117.
23. a 1 = 1, a 2 = x, an = 3a n – 1 + 6a n – 2
SOLUTION: The first five terms of the sequence are 1, x, 3x +6, 15x + 18 and 63x + 90.
Write a recursive formula for each sequence.
25. 32, 12, 7, 5.75, …
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The first five terms of the sequence are 1, x, 3x +6, 15x + 18 and 63x + 90.
10-5 Recursion and Iteration
Write a recursive formula for each sequence.
25. 32, 12, 7, 5.75, …
SOLUTION: 27. 1, 2, 9, 730, …
SOLUTION: 29. 480, 128, 40, 18, …
SOLUTION: 31. 68, 104, 176, 320, …
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Find the first three iterates of each function for the given initial value.
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10-5 Recursion and Iteration
31. 68, 104, 176, 320, …
SOLUTION: Find the first three iterates of each function for the given initial value.
33. f (x) = 12x + 8, x 0 = 4
SOLUTION: The first three iterates are 56, 680 and 8168.
35. f (x) = –6x + 3, x0 = 8
SOLUTION: The first three iterates are –45, 273 and –1635.
2
37. f (x) = –3x + 9, x 0 = 2
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SOLUTION: Page 7
first three
iterates
are –45, 273 and –1635.
10-5The
Recursion
and
Iteration
2
37. f (x) = –3x + 9, x 0 = 2
SOLUTION: The first three iterates are –3, –18 and –963.
2
39. f (x) = 2x – 5x + 1, x0 = 6
SOLUTION: The first three iterates are 43, 3484 and 24,259,093.
2
41. f (x) = x + 2x + 3,
SOLUTION: The first three iterates are 4.25, 29.5625 and 936.06640625.
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43. FRACTALS Consider the figures at the right. The number of blue triangles increases according to a specific
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first three
iterates
are 43, 3484 and 24,259,093.
10-5The
Recursion
and
Iteration
2
41. f (x) = x + 2x + 3,
SOLUTION: The first three iterates are 4.25, 29.5625 and 936.06640625.
43. FRACTALS Consider the figures at the right. The number of blue triangles increases according to a specific
pattern.
a. Write a recursive formula for the number of blue triangles in the sequence of figures.
b. How many blue triangles will be in the sixth figure?
SOLUTION: a. The number triangles in the figures are 1, 3 and 9.
Therefore, the recursive formula is
.
b.
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first three
iterates
are 4.25, 29.5625 and 936.06640625.
10-5The
Recursion
and
Iteration
43. FRACTALS Consider the figures at the right. The number of blue triangles increases according to a specific
pattern.
a. Write a recursive formula for the number of blue triangles in the sequence of figures.
b. How many blue triangles will be in the sixth figure?
SOLUTION: a. The number triangles in the figures are 1, 3 and 9.
Therefore, the recursive formula is
.
b.
45. CONSERVATION Suppose a lake is populated with 10,000 fish. A year later, 80% of the fish have died or been
caught, and the lake is replenished with 10,000 new fish. If the pattern continues, will the lake eventually run out of
fish? If not, will the population of the lake converge to any particular value? Explain.
SOLUTION: No; the population of fish will reach 12,500. Each year, 20% of 12,500 or 2500 fish, plus 10,000 additional fish, yields
12,500 fish.
47. SPREADSHEETS Consider the sequence with x0 = 20,000 and f (x) = 0.3x + 5000.
a. Enter x0 in cell A1 of your spreadsheet. Enter “= (0.3)*(A1) + 5000” in cell A2. What answer does it provide?
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b. Copy
cell
A2, highlight
cells A3 through A70,
and paste. What do you notice about the sequence?
c. How do spreadsheets help analyze recursive sequences?
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SOLUTION: No; the population of fish will reach 12,500. Each year, 20% of 12,500 or 2500 fish, plus 10,000 additional fish, yields
fish. and Iteration
10-512,500
Recursion
47. SPREADSHEETS Consider the sequence with x0 = 20,000 and f (x) = 0.3x + 5000.
a. Enter x0 in cell A1 of your spreadsheet. Enter “= (0.3)*(A1) + 5000” in cell A2. What answer does it provide?
b. Copy cell A2, highlight cells A3 through A70, and paste. What do you notice about the sequence?
c. How do spreadsheets help analyze recursive sequences?
SOLUTION: a. 11,000
b. It converges to 7142.857.
c. Sample answer: They make it easier to analyze recursive sequences because they can produce the first 100 terms
instantaneously; it would take a long time to calculate the terms by hand.
49. CCSS CRITIQUE Marcus and Armando are finding the first three iterates of f (x) = 5x – 3 for an initial value of
x0 = 4. Is either of them correct? Explain.
SOLUTION: Armando; Marcus included x0 with the iterates and only showed the first 2 iterates.
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b. It converges to 7142.857.
c. Sample answer: They make it easier to analyze recursive sequences because they can produce the first 100 terms
would take a long time to calculate the terms by hand.
10-5instantaneously;
Recursion anditIteration
49. CCSS CRITIQUE Marcus and Armando are finding the first three iterates of f (x) = 5x – 3 for an initial value of
x0 = 4. Is either of them correct? Explain.
SOLUTION: Armando; Marcus included x0 with the iterates and only showed the first 2 iterates.
51. REASONING Is the statement “If the first three terms of a sequence are identical, then the sequence is not
recursive” sometimes, always, or never true? Explain your reasoning.
SOLUTION: Sample answer: Sometimes; the recursive formula could involve the first three terms. For example, 2, 2, 2, 8, 20,… .
is recursive with a n + 3 = a n + a n + 1 + 2a n + 2.
53. WRITING IN MATH Why is it useful to represent a sequence with an explicit or recursive formula?
SOLUTION: Sample answer: In a recursive sequence, each term is determined by one or more of the previous terms. A recursive
formula is used to produce the terms of the recursive sequence.
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55. EXTENDED RESPONSE Bill launches a model rocket from ground level. The rocket’s height h in meters is given
2
by the equation h = –4.9t + 56t, where t is the time in seconds after the launch.
SOLUTION: Sample answer: In a recursive sequence, each term is determined by one or more of the previous terms. A recursive
10-5formula
Recursion
andtoIteration
is used
produce the terms of the recursive sequence.
55. EXTENDED RESPONSE Bill launches a model rocket from ground level. The rocket’s height h in meters is given
2
by the equation h = –4.9t + 56t, where t is the time in seconds after the launch.
a. What is the maximum height the rocket will reach? b. How long after it is launched will the rocket reach its maximum height? Round to the nearest tenth of a second.
c. How long after it is launched will the rocket land? Round to the nearest tenth of a second.
SOLUTION: a. Substitute 0 for h and find the vertex of the equation.
The vertex of a quadratic equation is
.
The vertex of the equation is (5.7, 160).
Therefore, the rocket will reach the maximum height of 160 m.
b. The vertex of the equation is (5.7, 160).
Therefore, it will take 5.7 s to reach the maximum height.
c. The rocket will land in 5.7 × 2 or 11.4 s after it is launched.
57. Which factors could represent the length times the width?
A (4x – 5y)(4x – 5y)
B (4x + 5y)(4x – 5y)
C (4x2 – 5y)(4x2 + 5y)
2
2
D (4x + 5y)(4x + 5y)
SOLUTION: Factor
the -area
of the
rectangle.
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b. The vertex of the equation is (5.7, 160).
Therefore, it will take 5.7 s to reach the maximum height.
The rocketand
willIteration
land in 5.7 × 2 or 11.4 s after it is launched.
10-5c.Recursion
57. Which factors could represent the length times the width?
A (4x – 5y)(4x – 5y)
B (4x + 5y)(4x – 5y)
C (4x2 – 5y)(4x2 + 5y)
2
2
D (4x + 5y)(4x + 5y)
SOLUTION: Factor the area of the rectangle.
Option C is the correct answer.
Write each repeating decimal as a fraction.
59. SOLUTION: The number
can be written as .
Find the value of r.
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C is the
answer.
10-5Option
Recursion
andcorrect
Iteration
Write each repeating decimal as a fraction.
59. SOLUTION: The number
can be written as .
Find the value of r.
Therefore:
61. SPORTS Adrahan is training for a marathon, about 26 miles. He begins by running 2 miles. Then, when he runs
every other day, he runs one and a half times the distance he ran the time before.
a. Write the first five terms of a sequence describing his training schedule.
b. When will he exceed 26 miles in one run?
c. When will he have run 100 total miles?
SOLUTION: a. This forms a geometric sequence.
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Given a 1 = 2, and r = 1.5
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c. When will he have run 100 total miles?
10-5SOLUTION: Recursion and Iteration
a. This forms a geometric sequence.
Given a 1 = 2, and r = 1.5
b. Given a n = 26
Find n.
He will exceed 26 miles in one run on the eighth session.
c. Given S n = 100
Find n.
He will have run 100 total miles during the ninth session.
State whether the events are independent or dependent.
63. choosing first and second place in an academic competition
SOLUTION: They are dependent events.
Find each product.
65. (x – 2)(x + 6)
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SOLUTION: They are dependent events.
10-5 Recursion and Iteration
Find each product.
65. (x – 2)(x + 6)
SOLUTION: 67. (4h + 5)(h + 7)
SOLUTION: 69. (2g + 7)(5g – 8)
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