Warm-up4/30/08

1
Warm-up 4/30/08

• Write the first six terms of the sequence with
  the given formula.
• a1 2
• an an 1 2n 1
• 2) an n2 1
• 3) What do you notice about your answers to
  Questions 1 and 2?

2

• Copy SLM for Unit 7 (chapter 4, 5)
• Disclaimer

3
Topic
Sequences and Series
Key Learning(s)
Use classic algorithms to find the sums of arithmetic and geometric series.
Unit Essential Question (UEQ)
How do you find the sums of arithmetic and geometric series?
4
Concept I
Formulas for Series
Lesson Essential Question (LEQ)
How do you find terms of sequences from explicit or recursive formulas? How do you find the limit of certain sequences?
Vocabulary
Sequence, explicit formula, recursive formula, arithmetic sequence, end behavior, Divergent, convergent, harmonic sequence, alternating harmonic sequence
5
Concept II
Arithmetic Series
Lesson Essential Question (LEQ)
How do you solve problems involving arithmetic series?
Vocabulary
Infinite series, finite series, arithmetic series,
6
Concept III
Geometric Series and Sequences
Lesson Essential Question (LEQ)
How do you solve problems involving geometric series? How do you solve problems involving infinite geometric series?
Vocabulary
Geometric series, infinite series,
7
Concept IV
Combinations
Lesson Essential Question (LEQ)
How do you use series to find combinations?
Vocabulary
combination
8
Concept V
Pascals Triangle and Binomial Theorem
Lesson Essential Question (LEQ)
How is Pascals Triangle used to expand polynomials?
Vocabulary
Pascals Triangle, Binomial Theorem, Binomial coefficients
9
8.1 Formulas for sequences

• LEQ How do you find terms of sequences from
  recursive or explicit formulas?
• Did you read? P. 488 - 493
• Sequence
• a function whose domain is the positive integers
• Explicit formula
• A formula in which you can find the nth term by
  plugging in any given integer n.
• Ex) Rn n(n1)

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• Recursive Formula
• Formula for a sequence in which the
• first term(s) is given and the nth terms is shown
  using all the preceding terms.
• Ex) a1 2
• an an 1 2n 1
• Try This
• What is the 9th term of the sequence 2, 4, 6, 8,
  ?
• Did you use an explicit formula or a recursive
  formula to get the 9th term?

11
Arithmetic Sequence

• Arithmetic Sequence
• The difference between the consecutive terms in
  the sequence is constant
• Ex) -7, -4, -1, 2, 5, 8
• General Formulas for Arithmetic Seq.
• Explicit an a1 (n 1)d
• Geometric a1
• an an 1 d, n gt1
• a1 is first term d is constant difference

12
Finding a position

• What position does 127 have in the arithmetic
  sequence below?
• 16,19,22,127
• a1 16
• d 3
• an 127
• 127 16 (n 1)3
• 127 3n 13
• N 38

13
Ex2)

• Which term is 344 in the arithmetic sequence
  8,15,22,29?
• a1 8
• d 7
• an 344
• 344 8 (n 1)7
• 344 7n 1
• n 49

14
Geometric Sequence

• Geometric Sequence
• The ratio of consecutive terms is constant.
• Ex) 3,3/2,3/4,3/8
• General Formulas for Geometric Seq.
• Explicit gn g1 r(n 1)
• Geometric g1
• gn rgn 1, n gt1
• g1 is first term r is constant ratio

15
Ex1)

• A particular car depreciates 25 in value each
  year. Suppose the original cost is 14,800.
• Find the value of the car in its second year.
• 25 is a rate of decrease year 2 75y1
• gn 14,800 (0.75)(2 1)
• gn 11,100

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• Write an explicit formula for the value of the
  car in its nth year.
• gn 14,800 (0.75)(n 1)
• In how many years will the car be worth about
  1000?
• 1,000 14,800 (0.75)(n 1)
• 0.065757 (0.75)(n 1)
• log0.065757 (n - 1)log(0.75)
• 9.36668 n 1
• N 10.3668

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Homework

• Worksheet 8.1
• Formulas for sequences
• 1 - 6

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Warm-up 5/1/08

• Given explicit formula Rn n(n 1)
• What is the 7th term of Rn?
• Find R30.
• If tn is a term in a sequence, what is the next
  term?

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• Go over 8.1 WS
• Finish 8.1 WS

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Calculator Tutorial

• Im learning with you!...
• http//education.ti.com/educationportal/sites/US/n
  onProductMulti/pd_onlinealgebra_free.html?bid2

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8.1 Assignment

• Section 8.1
• P.493
• 1-12, 13, 14, 19

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Warm-up 5/2/08

• Estimate the millionth term of each sequence to
  the nearest integer, if possible.
• The sequence defined by an 3n 2
• n 1
• for all positive integers n.
• The sequence defined by b1 400,
• bn 0.9n-1 for all integers gt 1.
• 3) The sequence defined by b1 6, bn 3/2bn-1
  for all integers gt 1.

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• CHECK 8.1
• ASSIGNMENT

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8.2 LIMITS OF SEQUENCES

• LEQ How do you find the limit of a sequence?
• Limit
• Defined as the value the function approaches the
  given value (8,- 8, 2, etc)
• Reading (10 minutes)
• p. 496 - 500

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• End Behavior
• What happens to a function f(n) as n gets very
  large (or small)
• Divergent Sequence
• A sequence that does not have a finite limit
• Ex) xn increase exponentially to 8
• Convergent Sequence
• A sequence that has a finite limit (gets close to
  a specific )
• Ex) The harmonic sequence approaches 0
• 1, ½, 1/3, ¼, 1/5, 1/6, 1/7.1/8 0

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Assignment

• 8.2 Worksheet

27
Warm-up 5/5/08

• Find the sum of the first 100 terms of the
  arithmetic sequence 3,7,11,
• Find the sum of the first 101 terms of this
  sequence.

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Interesting Facts

• Venus is the only planet that rotates clockwise.
• Jumbo jets use 4,000 gallons of fuel to take off
  .
• On average women can hear better than men.
• The MGM Grand Hotel of Las Vegas washes 15,000
  pillowcases per day!
• The moon is actually moving away from Earth at a
  rate of 1.5 inches per year.

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• In Australia, Burger King is called Hungry
  Jack's.
• Mosquitoes are attracted to the color blue twice
  as much as any other color.
• Jacksonville, Florida, has the largest total area
  of any city in the United States.
• The largest diamond ever found was an astounding
  3,106 carats!
• A comet's tail always points away from the sun.
• The lens of the eye continues to grow throughout
  a person's life.

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Check 8.2 Worksheet (HW)

1. -5
2. 56
3. 32
4. 7/4
5. Y 1
6. 1

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8.3 Arithmetic Series

• LEQ How do you solve problems involving
  arithmetic series?
• Main difference between a
• sequence and a series
• A sequence is a list of numbers.
• A series is the SUM of the sequence.

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• Infinite Series
• The number of things you add is infinite
• Ex) The sum of 1(n 1) from 0 to 8
• Finite Series
• The number of things you add is finite
• Ex) The sum of 1(n1) from 0 to 10
• Applied to Arithmetic Sequences
• An arithmetic sequence can be finite or infinite
  when it is the sum of terms in an arithmetic
  sequence.

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Ex1)

34
Ex2)

35
Ex3)

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Arithmetic Series Theorem

• The sum Sn a1 a an of an arithmetic
  series with first term a1 and constant difference
  d is given by
• (Final Term Known)
• Sn n/2(a1 an) or
• (Final Term Unknown)
• Sn n/2(2a1 (n 1)d)

37
Ex4)

• A student borrowed 4000 for college expenses.
  The loan was repaid over a 100-month period, with
  monthly payments as follows
• 60.00, 59.80, 59.60, ,40.20
• How much did the student pay over the life of the
  loan?
• Use Sn n/2(a1 an)
• Sn 100/2(60.00 40.20)
• Sn 5010

38
Ex5)

• A packer had to fill 100 boxes identically with
  machine tools. The shipper filled the first box
  in 13 minutes, but got faster by the same amount
  each time as time went on. If he filled the last
  box in 8 minutes, what was the total time that it
  took to fill the 100 boxes?
• Use Sn n/2(a1 an)
• S100 100/2(13 8)
• Sn 1050 min. or 17.5 hrs

39
Ex6)

• In training for a marathon, an athlete runs 7500
  meters on the first day, 8000 meters the next
  day, 8500 meters the third day, each day running
  500 meters more than on the previous day. How
  far will the athlete have run in all at the end
  of thirty days?
• Use Sn n/2(2a1 (n 1)d)
• S30 30/2(27500 (30 1)500)
• S30 442,500m or 442.5 km

40
Ex7)

• A new business decides to rank its 9 employees
  by how well they work and pay them amounts that
  are in arithmetic sequence with a constant
  difference of 500 a year. If the total amount
  paid the employees is to be 250,000, what will
  the employees make per year?
• Use Sn n/2(2a1 (n 1)d)
• 250000 9/2(2a1 (9 1)500)
• a1 25,778a9 29,778

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Practice

• 8.3 Worksheet
• Homework
• Section 8.3
• p. 507 508
• 3 7, 10 11, 13 - 15

42
Warm-up 5/6/08

1. Find a formula for the sum Sn of the first n
   terms of the geometric series 139
2. Use the formula to find the sum of the first 10
   terms of the series.

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8.3 Assignment Answers

• A series is a sum of the terms in a sequence.
• A. 35 B. 31
• A. 77 B. 65
• 500,500
• A. 7372.50
• B. 1372.50

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• -4
• 873,612
• 78
• 19 rows, 10 left over
• 21

45

• There are geometric and arithmetic sequences
• There are also geometric and arithmetic series.
• A geometric series is the sum of the terms in a
  geometric sequence.

46
Theorem

• The sum of the finite geometric sequence with
  first term g1 and constant ratio r ? 1 is given
  by
• Sn g1(1 rn)
• 1 r
• For finite 0 lt r lt 1
• The proof for the formula can be seen on pg. 510
  of the textbook.

47
Equivalent Formula

• If the rate (r) is gt 1, another formula can be
  used (this would be an infinite series).
• Sn g1(rn 1)
• r - 1

48
Ex1)

• Find the sum of the first six terms of the
  geometric sequence
• 10(0.75)(i 1)
• 32.88085938
• 10(0.75) (1 1) 10(0.75) (2 1) 10(0.75)
  (3 1) 10(0.75) (4 1) 10(0.75) (5 1)
  10(0.75) (6 1)

49
Ex2)

• In a set of 10 Russian nesting dolls, each doll
  is 5/6 the height of the taller one. If the
  height of the first doll is 15 cm, what is the
  total height of the doll?
• Sn g1(1 rn)
• 1 r
• Sn 15(1 (5/6)10)
• 1 (5/6)
• 75 cm

50
Ex3)

• Suppose you have two children who marry and each
  of them has two children. Each of these
  offspring has two children, and so on. If all of
  these progeny marry but non marry each other, and
  all have two children, in how many generations
  will you have a thousand descendants? Count your
  children as Generation 1.
• 1000 2(2n 1)
• 2 1

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Practice

• 8.4 Worksheet

52
Assignment

• Section 8.4
• p. 512 -513
• 5 7, 10 (see Ex3), 11, 13,14, 18 - 20

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Warm-up 5/7/08

• Write the first six terms of the geometric
  sequence with first term -2 and constant ratio 3.
• -2,-6,-18,-54,-162,-486
• Find the sum of the first six terms for 1.
• -728

54
Interesting Facts

• Flamingos can only eat with their heads upside
  down.
• Babies start dreaming even before they're born.
• The word 'gymnasium' comes from the Greek word
  gymnazein which means 'to exercise naked.'
• 4.5 pounds of sunlight strike the Earth each day.
• 40 degrees Celsius is equal to -40 degrees
  Fahrenheit. Your brain is 80 water.

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• Your brain is 80 water.
• The phrase 'rule of thumb' is derived from and
  old English law which stated that you couldn't
  beat your wife with anything wider than your
  thumb.
• It is illegal to mispronounce 'Arkansas' while in
  the state of Arkansas!
• There are more than 1,000 chemicals in a cup of
  coffee. Of these, only 26 have been tested, and
  half caused cancer in rats.
• The Pittsburgh Steelers were originally called
  the Pirates.
• Over 98 percent of Japanese people are cremated
  after they die.
• The penguin is the only bird that can swim, but
  cannot fly.

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8.4p. 512 -513 5 7, 10, 11, 13,14, 18 - 20

• 5.98
• 66,485.13
• Not the million
• 12 3
• 17 terms 127.037831
• 4,265.625

57

• -33.25
• 18)(2i 1) is gt by 20
• a) 25,250 b) 218.750
• 2,4/3,8/7,16/15,32/31
• b) yes 1

58
Questions?

• Quiz over 8.1 - 8.3
• 20 minutes
• Then, read pg.
• 516 - 520

59
Exploring Infinite Series

• In class activity
• p. 515

60
8.5 Infinite Series

• How do you solve problems involving infinite
  geometric series?
• What would an infinite series be?
• Recall
• Divergent
• Convergent

61
Simply Put

• With arithmetic series, you have to add some
  terms together to determine whether it appears to
  be divergent or convergent
• (no good method covered in this class)
• With geometric series, if rlt 1, the series
  converges formula S8 g1
• 1 r
• If r gt 1 the series diverges

62
Practice

• 8.5 Worksheet

63
Warm-up 5/8/08

• Write the first five terms of the harmonic
  series.
• Use a calculator to find how many terms of the
  series must be added for the sum to exceed 3.
• Use a calculator to find how many terms of the
  series must be added for the sum to exceed 5.
• T/F The harmonic series is divergent.

64
Did you know

• Persia changed its name to Iran in 1935.
• Rice flour was used to strengthen some of the
  bricks that make up the Great Wall of China.
• Russia is the world's largest country with an
  area of 17,075,400 square kilometers.
• Seven asteroids were especially named for the
  Challenger astronauts who were killed in the 1986
  failed launch of the space shuttle.
• Soil that is heated by geysers is now making it
  possible to produce bananas in Iceland.
• Some asteroids have other asteroids orbiting
  them.
• St. Paul, Minnesota was originally called Pigs
  Eye after a man named Pierre "Pig's Eye" Parrant
  who set up the first business there.

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• Stalks of sugar cane can reach up to 30 feet.
• Tasmania is said to have the cleanest air in the
  world.
• Thailand used to be called Siam.
• The Amazon rainforest produces more than 20 the
  world's oxygen supply.
• The Angel Falls in Venezuela were named after an
  American pilot, Jimmy Angel, whose plane got
  stuck on top of the mountain while searching for
  gold.
• The Apollo 17 crew were the last men on the moon.
  
• The Chihuahua Desert is the largest desert in
  North America, and is over 200,000 square miles.
• The Dead Sea has been sinking for the last
  several years.

66

• Pass back papers
• Finish 8.5 Worksheet

67

• In Class,
• Complete
• Self Test
• p. 550 1 - 11

68
HW

• Chapter Review
• p. 551
• 1 14, 21, 22, 24 - 26, 27 36, 42 - 46

69
Warm-up 5/9/08

• Evaluate the arithmetic or geometric sequence
  given
• 103 120 137 154 290
• 2358
• The sum of the first 100 terms of the sequence
  (4k 13).
• 18,900
• The sum of the first 20 terms of the sequence
  10(0.6)n 1
• 24.999

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Interesting Facts

• In 2001, St. Patrick's Day was banned in Ireland
  because of the scare caused by foot and mouth
  disease.
• A 13-year-old boy in India produced winged
  beetles in his urine after hatching the eggs in
  his body.
• Airports that are at higher altitudes require a
  longer airstrip due to lower air density.
• Amish people do not believe in the use of aerosol
  air fresheners.
• Annually 17 tons of gold is used to make wedding
  rings in the United States.
• Approximately 1 billion stamps are produced in
  Australia annually.

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• Being unmarried can shorten a man's life by ten
  years.
• DC-10, the name of an airplane stands for
  "Douglas Commercial."
• Every U.S. bill regardless of denomination costs
  just 4 cents to make.
• Fires on land generally move faster uphill than
  downhill.
• If someone was to fly once around the surface of
  the moon, it would be equal to a round trip from
  New York to London.
• In 1907, on New Year's Eve, the original ball
  that was lowered in Times Square was made of wood
  and iron and had 100 light bulbs on it.

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• Approximately 75 of human poop is made of water.
• It has been estimated that the fear of the number
  13 costs Americans more than 1 billion per year!
• Smokers eat more sugar than non-smokers do.
• Beavers can swim half a mile underwater on one
  gulp of air.
• It takes twelve ears of corn to make a tablespoon
  of corn oil.
• 10 of the tributaries flowing into the Amazon
  river are as big as the Mississippi river.

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Reminders

• All library books are due by the end of today.
• Check your lockers, etc.
• Your final 5/16 (next Friday)

74

• Questions?
• Collect Chapter 8 Review (E.C.)
• Chapter 8 Test
• Teacher Evaluations

75
Agenda this week

• Mon Thurs Review/Mini-Projects
• (if you are going to exempt the final, all work
  must be turned in!)
• Friday Final
• Today Return Ch. 8 Tests go over
• Begin Chapter 1 Mini-Project

76
Random Facts

• Baby beavers are called kittens.
• You have no sense of smell when you're sleeping!
• Ants dont sleep.
• An albatross can sleep while it flies!
• The earth is .02 degrees hotter during a full
  moon.
• By feeding hens certain dyes they can be made to
  lay eggs with multi-colored yolks.

77

• 40 of all indigestion remedies sold in the world
  are bought by Americans.
• Animals will not eat another animal that has been
  hit by a lightning strike!
• Dragonflies can travel up to 60 mph.
• The average 1 1/4 lb. lobster is 7 to 9 years
  old.
• Until President Kennedy was killed, it wasnt a
  federal crime to assassinate the President.
• Each year, 24,000 Americans are bitten by rats!

78

• Go over Chapter 8 Test
• Chapter 1 Test Form D

79
Warm-up 5/13/08

• The following gives the number of World Wide
  Websites during a period of years.
• Make a scatter plot.
• Find a good model (linear, quadratic, etc)

Months since 1/1/1993 www Sites
6 130
12 623
18 2,738
24 10,022
30 23,500
37 100,000
42 230,000
80
Interesting Facts

• Crushed cockroaches can be applied to a stinging
  wound to help relieve the pain.
• The average human body contains enough iron to
  make a small nail.
• Astronauts cannot burp in space.
• A mole can dig a hole 300 feet deep in one night.
• The sting from a killer bee contains less venom
  than the sting from a regular bee!

81

• A rat can go without water longer than a camel
  can.
• Cats cannot taste sweet things.
• A male baboon can kill a leopard.
• In its ancient form, the carrot was purple, not
  orange.
• There are more fatal car accidents in July than
  any other month.
• About 1 in 30 people, in the U.S., are in jail,
  on probation, or on parole!
• Approximately 70,000 people in the U.S. are both
  blind and deaf!

82
Instructions for the next week

• Some questions that are addressed
• Do seniors have to be at school if they are
  exempting exams?
• Seniors exempting either 1st or 2nd block exams
  will be allowed excused absences in the
  applicable class on both Thursday and Friday.

83

• What if I have seniors and underclassmen in the
  same class?
• There will be two versions of your final exam. 
  When seniors take the exam, let your
  underclassmen also take it.  Use it as part of
  your review before giving underclassmen the
  second version of your exam next week.

84

• What if a senior is not exempt from the exam and
  is absent the day of the exam?
• If a senior is absent for a Friday exam, hell
  have to make it up on Monday (a second version of
  the exam).
• If a senior is absent for both days of senior
  exams, theyll have to take the exams on Tuesday
  and Wednesday with the underclassmen.

85

• Will the senior get a chance to retake an exam if
  the grade he received on his exam causes the
  grade in the class to drop below passing? 
• The senior will be allowed one retake of a final
  exam (a second version) on Tuesday, May 20 only
  if the senior comes in to meet with you on
  Monday, May 19 to go over the first exam taken. 
• You decide on the time for review and retake.

86

• ALL grades for ALL seniors should be in Power
  Grade no later than 1200 noon on Monday (with
  the few exceptions resulting from 3 or 4
  above). 

87

• NO PARTIES and NO FREE DAYS.

88

• Seniors have limited activities next week.  Only
  seniors who are taking final exams should be in
  the building (i.e., its not time for them to
  hang out in your class because theyre done
  with high school).  After each activity, seniors
  will be excused from school for the remainder of
  the day. 

89

• Monday, May 19 Fun photo day, 900 AM.  Some
  group shots will be taken in and possibly around
  the stadium.
• Tuesday, May 20 Graduation Practice, 830 AM,
  Roquemore Field
• Wednesday, May 21 Graduation Practice, 830 AM,
  Roquemore Field
• Thursday, May 22 Senior Breakfast (optional,
  RSVP to senior homeroom teacher by Monday, May
  19).

90
Plan for Algebra III

• Project Folders
• Four Projects Total
• Turned in by your last day (if youre a senior
  exempting my exam, that will be tomorrow!)
• If youre a senior and taking the final exam, I
  will give you a study guide tomorrow

91
Project Folder Expectations

• They should be neat
• All problems should be solved to the best of your
  ability
• Any graphs graphic representations should be
  complete and appropriate
• All work should be included (consider doing one
  problem per page)
• All parts should be CLEARLY labeled
• The final folder will count as a test grade