airithmatic 10, 15, 20, 25, 30 constant slope · 2018-10-05 · in that sequence. the constant...

AdvAlg37RecursiveFormulaArithmeticSeq.notebook

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October 05, 2018

Jul 98:59 PM

Recursive Formulas for Arithmetic SequencesNotes 37

Remember: A Recursive Sequence

Write the first five terms of the sequence _______________

What is the difference between the terms?

This _______________ difference is the _____________.

First term of the sequence

Rule to find each term after the 1st term

Constant Slope

10, 15, 20, 25, 30

This is a notation for the previous term

for integers n ≥ 2

airithmatic


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1. What is an arithmetic sequence?

2. In an arithmetic sequence what is true about the difference between any term and the preceding term?

3. Define the recursive definition for an arithmetic sequence.

A list of numbers in which the order matters and which has a constant difference between successive terms?

It will be the same for every adjacent pair in that sequence. The constant difference.

It states the first term and a formula for finding the nth term from the preceding(n1) term by adding the constant difference.

We start n at 2 because we explicitly state the 1st terms(we already know the first term)

Thinking recursively: requires the calculation of a term using the previous term.


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The 1st term is 53 and we find all other terms by subtracting 7 from the previous term


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or


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8. When working with an arithmetic sequence the first term is referred toby what phrase?

9. An arithmetic sequence represents a constant increasing or constant decreasing situation.

a. What is the other name given to an arithmetic sequence?

b. What does this tell us about all the points generated by the sequence?

The initial condition

Linear sequence

When graphed all the points will form a line. This will be a discrete graph which means it contains only points not a line connecting them.


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Arithmetic Sequence: A sequence with a ______________

________________ between consecutive terms. Also called

a _________________ sequence.

Theorem The sequence defined by the recursive formula

Constantdifferencelinear

integers n > 2

is the _____________ sequence with first term ____ and constant difference __.arithmetic a1 d


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Write a recursive formula for the arithmetic sequence

17, 28, 39, 50, ………….

Find constant difference_______

Is 10, 8, 7, 3, 20 an arithmetic sequence? Why or why not?


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Sep 28:30 AM

Write a recursive formula for the sequence 150,75,37.5,18.75....


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Sep 28:33 AM

Due to an increasing population, the town of Valley Heights is concerned about its water supply. The town council has voted to immediately add 20,000 acrefeet of water to its reservoir capacity of 3 million acrefeet and to add an additional 20,000 acrefeet of water each year. Write a recursive formula to express the capacity of the reservoir in n years.


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Write the first five terms of each sequence and write a recursive formula for each.

An arithmetic sequence has first term 4 and constant difference 20.

An arithmetic sequence has first term 0.3 and constant difference 0.1


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Sep 28:39 AM

Hector has an empty jar. Every workday he plans to add the 30 cents change from his bus fare.

A. At the start, how much change is in the jar ?

B. Write a recursive formula that gives the amount of change in the jar after

the nth workday.


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WS 3.7 & 3.8.pdf


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WS 36 to 38 B.pdf


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WS 36 to 38 B KEY.pdf


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Attachments

WS 3.7 & 3.8.pdf

WS 36 to 38 B.pdf

WS 36 to 38 B KEY.pdf

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Name

L E S S O NM A S T E R

Vocabulary

1. In your own words, define arithmetic sequence.

Skills Objective D

2. Use the arithmetic sequence 0.5, 0.75, 1.00, 1.25, . . . .

a. Describe this sequence in words.

b. Write a recursive definitionfor this sequence.

3. An arithmetic sequence has first term 6 and constant difference 4.

a. Write the first 5 terms of the sequence.

b. Write a recursive definition forthe sequence.

Properties Objective F

4. A sequence is defined recursively as for integers n ≥ 2.

a. Find the first 7 terms of this sequence.

b. Is the sequence arithmetic? Justify your answer.

5. Is the sequence 9, 27, 81, 243, . . . arithmetic? Justify your answer.

Uses Objective G

6. Pak bought a pound of coffee beans. Each morning

she uses ounce to brew coffee.

a. How many ounces of coffee beans doesshe have left after the first morning?

b. Write a recursive definition for the amount of coffee beans left after n mornings.

34

a15 12

an5 a

n 2 12 3,

Questions on SPUR ObjectivesSee pages 197-201 for objectives.3-7

A

Name


Skills Objective D

In 1 and 2, an arithmetic sequence is given.a. Write a formula for the nth term. b. Find a

200.

1. 19, 25, 31, 37, . . . a. b.

2. -4, -6.5, -9, -11.5, . . . a. b.

In 3 and 4, a recursive definition for a sequence isgiven. Write an explicit formula for the sequence.

3. 4.

5. Write a recursive definitionfor the sequence defined explicitly by a

n5 9n 2 7.

6. An arithmetic sequence has a3

5 11.1 and a7

5 23.9.

a. Write an explicit formula for the sequence.

b. Write a recursive definitionfor the sequence.

7. Find the 250th term of the linear sequence 5p, 8p, 11p, 14p, . . .

Properties Objective F

In 8–10, determine whether or not the given formuladescribes an arithmetic sequence. Justify your answer.

8. an5 n3 2 6

9. bn5 4n 1 7

10. cn5 n 2

Uses Objective G

11. A TV shopping club that had 1218 gold necklaces for $125each sold 42 necklaces each minute the item was featured.

a. Write an explicit formula that gives thenumber of necklaces left a

nafter n minutes.

b. How many minutes does this item need to be featured before the club would sell out?

53

23

d15 π

dn5 d

n 2 11 2π,

a15

3–5

an5 a

n 2 11

2–5,

Questions on SPUR ObjectivesSee pages 197-201 for objectives.3-8

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SMART Notebook

Name


Vocabulary

Questions on SPUR Objectives3-6B

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1. a. What is the range of r, the correlationcoefficient for a set of data?

b. Suppose r 5 -0.9 for a line fit to a set of data. What does this tell you about the strength of the linear relationshipbetween the variables?

Uses Objective J: Fit lines to data.

2. The following data give the number of city-councilmembers in six cities with various populations.

a. Draw a scatterplot ofthe data.

b. Find an equation of the regression line.

c. Graph the regression lineon your scatterplot.

d. Use your equation to predict the numbers of city-councilmembers in a city with apopulation of 250,000.

e. Interpret the strength of the linear relationship based onthe correlation coefficient.

Population 45,000 16,000 320,000 108,000 61,000 176,000

City-Council8 7 24 19 12 15

Members

x100

y

10

20

30

20 30Population (10,000s)

Nu

mb

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of

Me

mb

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3. The following data give the number of FrosteeTreets sold and the high temperature on 10 differentsummer days.


b. Find an equation of theregression line.

c. Graph the regression line onyour scatterplot.

d. What does the slope in yourequation mean in terms ofthis situation?

e. Use your equation to predict the number of FrosteeTreets that would be sold on a 95˚ day. How close is thisvalue to the actual data?

f. Does this situation or the one described in Item 2 exhibitthe stronger linear relationship? How do you know this?

4. Use the regression line to determine whether (3.2, 4.08),(4.5, 4.925), and (6, 5.92) lie on a line. Explain your answer.

Name

© LESSON MASTER 3-6B page 2

Temperature 88˚ 71˚ 84˚ 98˚ 95˚ 88˚ 80˚ 72˚ 77˚ 85˚

Frostee Treets2,044 1,099 1,941 2,708 2,539 1,886 1,522 503 1,493 1,216

Sold

t70

F

10

20

30

80 90Temperature (°F)

Tre

ets

So

ld (

100s

)

100

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Vocabulary


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1. Give an example of an arithmetic sequence. Then explainwhy it is an arithmetic sequence.

Skills Objective D: Evaluate or find recursive formulas for arithmetic sequences.

In 2–4, an arithmetic sequence is given.a. Describe the sequence in words.b. Write a recursive formula for the sequence.

2. 17, 28, 39, 50, . . .

a. b.

3. 80, -160, -400, -640, . . .

a. b.

4. , , 1, , . . .

a. b.

In 5 and 6, an arithmetic sequence is described.a. Write the first five terms of the sequence.b. Write a recursive formula for the sequence.

5. An arithmetic sequence has first term4 and constant difference 20.

a. b.

6. An arithmetic sequence has first term0.3 and constant difference -0.1.

a. b.

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Properties Objective F: Recognize properties of arithmeticsequences.

In 7 and 8, tell whether or not the sequence could be anarithmetic sequence. Justify your answer.

7. 400, 200, 100, 50, . . .

8. 49, 44, 39, 34, . . .

Uses Objective G: Model situations involving arithmetic sequences.

9. Mrs. Machado contributed $50 to a local charity and pledgedto donate another $5 every month thereafter.

a. Write a sequence that shows Mrs. Machado’s totalcontributions during the first six months.

b. Write a recursive formula for the sequencein Part a.

10. Hector has an empty jar. Every workday he plansto add the 30¢ change from his bus fare.

a. At the start, how much change is in the jar?

b. Write a recursive formula that gives the amountof change in the jar after the nth workday.

11. One afternoon, Matt bought an 85-ounce box of dishwasher detergent. Each morning he uses 3 ounces.

a. Write a recursive formula for a sequence thatgives the amount of detergent Matt will have left on the nth evening.

b. Find the amount of detergent left on the10th evening.

Name


Name


Skills Objective D: Evaluate or find explicit formulas forarithmetic sequences.


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In 1–4, an arithmetic sequence is given.a. Find a formula for the nth term.b. Find a

25.

1. -22, -19, -16, -13, . . . 2. , , , , . . .

a. a.

b. b.

3. 3.2, 4.9, 6.6, 8.3, . . . 4. 75, 25, -25, -75, . . .

a. a.

b. b.

In 5 and 6, a recursive formula for a sequence is given.Write an explicit formula for the sequence.

5.

6.

7. Write a recursive formula for the sequence defined explicitly by c

n5 45 1 (n 2 1)5.

8. Write a recursive formula for the sequence defined explicitly by d

n5 300n 1 50.

In 9 and 10, two terms of an arithmetic sequence are given.a. Write an explicit formula for the sequence.b. Write a recursive formula for the sequence.

9. s25 12 and s

85 60 10. a

55 1.6 and a

105 -0.9

a. a.

b. b.

b15 5.75

bn5 b

n 2 12 1.25, for n ≥ 2.

a15 20

an5 a

n 2 11 12, for n ≥ 2.

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11. Find the 150th term of the arithmetic sequence7g, 11g, 15g, 19g, . . .



12. an5 9n 1 18

13. tn5 3n2 1 4

14. un5 n 2 1

15. vn5 n 2 30


16. A wading pool, filled to a depth of 36 inches, drainsat the rate of about 3 inches per hour.

a. Write an explicit formula that gives thedepth of the water after n hours.

b. How deep will the water be after 6 hours?

17. A vehicle emissions test center tests 320 vehiclesevery weekday.

a. Write an explicit formula that gives thetotal number of cars tested after n days.

b. How many weeks will it take to test240,000 cars?

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Vocabulary


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1. Give the general form of a point-slope equation for a line.

y 2 y1

5m(x 2 x1)

Skills Objective B: Find an equation for a line given two points on itor given a point on it and its slope.

In 2–13, find an equation for the line with the giveninformation. Write your equation in standard form withintegers for A, B, and C.

2. slope -1, through (4, -3)

3. slope , through (6, 1)

4. through (1, 4) and (-2, -2)

5. through (1, 8) and (9, 8)

6. slope -4, y-intercept 6

7. slope 3, x-intercept -7

8. through (-3, 2) and (-3, 0)

9. slope -3, through (0, 0)

10. x-intercept 2, y-intercept 5

11. through (-4, 1) parallel to 4x 1 2y 5 7

12. through (6, 6) with undefined slope.

13. x-intercept 12, parallel to x 2 6y 5 10

Properties Objective E: Recognize properties of linear functions.

14. Fill in the three blanks with the correct values. According tothe Point-Slope Theorem, the line containing (4, -5) withslope 2 has equation y 2 5 (x2 ).

15. True or false. The y-axis has a slope of zero.Justify your answer.

False; sample: y-axis is vertical, so its slope

is undefined.

42-5

???

54

Samples aregiven.

x 1 y 5 1

5x 2 4y 5 26

2x 2 y 5 -2

y 5 8

4x 1 y 5 6

3x 2 y 5 -21

x 5 -3

3x 1 y 5 0

5x 1 2y 5 10

2x 1 y 5 -7

x 5 6

x 2 6y 5 12

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Uses Objective I: In a real-world context, find an equation for a linecontaining two points.

16. Card Carriers charges $36 to print 1,200 business cards and$56 for 2,700 cards. Assume the relationship between theprice and the number of business cards is linear.

a. Write an equation giving price as a functionof the number of cards printed.

b. Find the set-up cost (the cost for printing0 cards).

c. Find the cost of printing 6,000 cards.

17. Last week, Mr. Chinn sold $20,000 worth of newspaperadvertisements and earned $800. The week before, he sold$26,000 worth of advertisements and earned $860. Assumethe relationship between Mr. Chinn’s weekly earnings and thevalue of the advertisements he sells is linear.

a. Write an equation giving Mr. Chinn’s weeklyearnings as a function of the value of theadvertisements he sells.

b. In this situation, what do the slope and y-intercept mean?

Slope is rate of commission, 1%. y-intercept

is base salary, without commission.

c. If in one week Mr. Chinn sells $30,000 worthof advertisements, how much will he earn?

Uses Objective K: Model situations leading to piecewise-linear functions.

18. Northstreet Disposal Company charges $30 to send out atruck to pick up debris. For the first 5 cubic yards of rubbish,the company charges an additional $10 per cubic yard. Foreach additional cubic yard, to a maximum of 45 cubic yards,the company charges $6.

a. What is the cost to have the following amount of rubbish removed?

3 cu yd 6 cu yd

30 cu yd 45 cu yd

b. Write an equation that gives the cost c for picking up ycubic yards of rubbish for the following values of y.

0 ≤ y ≤ 5 5 ≤ y ≤ 45c 5 6y 1 50c 5 10y 1 30

$320$230

$86$60

Name


y 5 x 1 20

$20$100

y 5 .01x 1 600

$900

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Vocabulary


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1. a. What is the range of r, the correlationcoefficient for a set of data?

b. Suppose r 5 -0.9 for a line fit to a set of data. What does this tell you about the strength of the linear relationshipbetween the variables?

Sample: There is a strong linear

relationship.

Uses Objective J: Fit lines to data.

2. The following data give the number of city-councilmembers in six cities with various populations.


b. Find an equation of the regression line.

y 5 .000052x1 7.9

c. Graph the regression lineon your scatterplot.

d. Use your equation to predict the numbers of city-councilmembers in a city with apopulation of 250,000.

21 members

e. Interpret the strength of the linear relationship based onthe correlation coefficient.

Sample: It is reasonably strong,

since r .89.

-1 ≤ r ≤ 1

Sampleequationis given for b.

Population 45,000 16,000 320,000 108,000 61,000 176,000

City-Council8 7 24 19 12 15

Members

x100

y

10

20

30

20 30Population (10,000s)

Nu

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3. The following data give the number of FrosteeTreets sold and the high temperature on 10 differentsummer days.


b. Find an equation of theregression line.

y 5 67.8x2 3984

c. Graph the regression line onyour scatterplot.

d. What does the slope in yourequation mean in terms ofthis situation?

Sample: Each increase in temperature of

1̊ results in the sale of 68 Frostee Treets.

e. Use your equation to predict the number of FrosteeTreets that would be sold on a 95˚ day. How close is thisvalue to the actual data?

Sample: 2457 Frostee Treets; it is close,

within about 3%.

f. Does this situation or the one described in Item 2 exhibitthe stronger linear relationship? How do you know this?

Sample: this situation has the stronger

linear relationship, since |r | is closer to

1(r .91).

4. Use the regression line to determine whether (3.2, 4.08),(4.5, 4.925), and (6, 5.92) lie on a line. Explain your answer.

No; r .9998 ≠ 1

Name


Sample isgiven for b.

Temperature 88˚ 71˚ 84˚ 98˚ 95˚ 88˚ 80˚ 72˚ 77˚ 85˚

Frostee Treets2,044 1,099 1,941 2,708 2,539 1,886 1,522 503 1,493 1,216

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10

20

30

80 90Temperature (°F)

Tre

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100s

)

100

259

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Vocabulary


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1. Give an example of an arithmetic sequence. Then explainwhy it is an arithmetic sequence.

Sample: 15, 20, 25, 30, . . . ; There is a constant

difference, 5, between consecutive terms.

Skills Objective D: Evaluate or find recursive formulas for arithmetic sequences.

In 2–4, an arithmetic sequence is given.a. Describe the sequence in words.b. Write a recursive formula for the sequence.

2. 17, 28, 39, 50, . . .

a. b.Arithmetic sequence with

first term 17, constant

difference 11

3. 80, -160, -400, -640, . . .


first term 80, constant

difference -240

4. , , 1, , . . .


first term , constant

difference

In 5 and 6, an arithmetic sequence is described.a. Write the first five terms of the sequence.b. Write a recursive formula for the sequence.

5. An arithmetic sequence has first term4 and constant difference 20.

a. b.4, 24, 44, 64, 84

6. An arithmetic sequence has first term0.3 and constant difference -0.1.

a. b.0.3, 0.2, 0.1, 0, -0.1

13

13

43

23

13

a1

5 17a

n5 a

n 2 1111,

for n ≥ 2.

a1

5 80a

n5 a

n 2 12240,

for n ≥ 2.

a1

5 4a

n5 a

n 2 1120,

for n ≥ 2.

a1

51–3,

an

5 an 2 1

11–3,

for n ≥ 2.

a1

5 0.3a

n5 a

n 2 120.1,

for n ≥ 2.

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In 7 and 8, tell whether or not the sequence could be anarithmetic sequence. Justify your answer.

7. 400, 200, 100, 50, . . .

No; the difference is not constant.

8. 49, 44, 39, 34, . . .

Yes; there is a constant difference of -5.


9. Mrs. Machado contributed $50 to a local charity and pledgedto donate another $5 every month thereafter.

a. Write a sequence that shows Mrs. Machado’s totalcontributions during the first six months.

$50, $55, $60, $65, $70, $75

b. Write a recursive formula for the sequencein Part a.

10. Hector has an empty jar. Every workday he plansto add the 30¢ change from his bus fare.

a. At the start, how much change is in the jar?

b. Write a recursive formula that gives the amountof change in the jar after the nth workday.

11. One afternoon, Matt bought an 85-ounce box of dishwasher detergent. Each morning he uses 3 ounces.

a. Write a recursive formula for a sequence thatgives the amount of detergent Matt will have left on the nth evening.

b. Find the amount of detergent left on the10th evening. 58 ounces

Name


a1

5 50a

n5 a

n 2 115,

for n ≥ 2.

0¢a

15 0

an

5 an 2 1

130,for n ≥ 2.

a1

5 85a

n5 a

n 2 123,

for n ≥ 2.

Name


Skills Objective D: Evaluate or find explicit formulas forarithmetic sequences.


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In 1–4, an arithmetic sequence is given.a. Find a formula for the nth term.b. Find a

25.

1. -22, -19, -16, -13, . . . 2. , , , , . . .

a. a.

b. b.

3. 3.2, 4.9, 6.6, 8.3, . . . 4. 75, 25, -25, -75, . . .

a. a.

b. b.

In 5 and 6, a recursive formula for a sequence is given.Write an explicit formula for the sequence.

5.

6.

7. Write a recursive formula for the sequence defined explicitly by c

n5 45 1 (n 2 1)5.

8. Write a recursive formula for the sequence defined explicitly by d

n5 300n 1 50.

In 9 and 10, two terms of an arithmetic sequence are given.a. Write an explicit formula for the sequence.b. Write a recursive formula for the sequence.

9. s25 12 and s

85 60 10. a

55 1.6 and a

105 -0.9

a. a.

b. b.

an

5 -.5n 1 4.1sn

5 8n 2 4

b15 5.75

bn5 b

n 2 12 1.25, for n ≥ 2.

a15 20

an5 a

n 2 11 12, for n ≥ 2.

-112544

an

5 -50n 1 125an

5 1.7n 1 1.5

50

an

51–2n 1

5–4an 5 3n 2 25

134

114

94

74

an

5 12n 1 8

bn5 -1.25n 1 7

c1

5 45c

n5 c

n 2 115,

for n ≥ 2.

d1

5 350d

n5 d

n 2 11 300,

for n ≥ 2.

s1

5 4s

n5 s

n 2 118,

for n ≥ 2.

a1

5 3.6a

n5 a

n 2 12.5,

for n ≥ 2.

554

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11. Find the 150th term of the arithmetic sequence7g, 11g, 15g, 19g, . . .



12. an5 9n 1 18

Yes; there is a constant difference of 9.

13. tn5 3n2 1 4

No; there is not a constant difference.

14. un5 n 2 1

Yes; there is a constant difference of .

15. vn5 n 2 30

Yes; there is a constant difference of -1.


16. A wading pool, filled to a depth of 36 inches, drainsat the rate of about 3 inches per hour.

a. Write an explicit formula that gives thedepth of the water after n hours.

b. How deep will the water be after 6 hours?

17. A vehicle emissions test center tests 320 vehiclesevery weekday.

a. Write an explicit formula that gives thetotal number of cars tested after n days.

b. How many weeks will it take to test240,000 cars?

12

Name


603g

12

an

5 -3n 1 36

18 inches

an

5 320n

150 weeks

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