Warm-up 5.1 by wanghonghx


									 Warm-up                         4/30/08
Write the first six terms of the sequence with
   the given formula.

1) a1 = 2
   an = an – 1 + 2n – 1
2) an = n2 + 1
3) What do you notice about your answers
   to Questions 1 and 2?
Copy SLM for Unit 7
   (chapter 4, 5)

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?
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?
Sequence, explicit formula, recursive formula,
arithmetic sequence, end behavior,
Divergent, convergent, harmonic sequence,
alternating harmonic sequence
Concept II
Arithmetic Series
Lesson Essential Question (LEQ):
How do you solve problems involving
arithmetic series?
Infinite series, finite series, arithmetic
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?
Geometric series, infinite series,
Concept IV
Lesson Essential Question (LEQ):
How do you use series to find
Concept V
Pascal’s Triangle and Binomial
Lesson Essential Question (LEQ):
How is Pascal’s Triangle used to
expand polynomials?
Pascal’s Triangle, Binomial Theorem,
Binomial coefficients
  §8.1: Formulas for sequences
LEQ: How do you find terms of sequences from
           recursive or explicit formulas?
            Did you read? P. 488 - 493
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(n+1)
                  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:
1) What is the 9th term of the sequence 2, 4, 6, 8,
2) Did you use an explicit formula or a recursive
    formula to get the 9th term?
        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 >1
a1 is first term      d is constant difference
     Finding a position
What position does 127 have in the
   arithmetic sequence below?
              a1 = 16
             an = 127
       127 = 16 + (n – 1)3
          127 = 3n + 13
              N = 38
Which term is 344 in the arithmetic
sequence 8,15,22,29…?
                 a1 = 8
                an = 344
          344 = 8 + (n – 1)7
              344 = 7n + 1
                 n = 49
       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 >1
g1 is first term       r is constant ratio
  A particular car depreciates 25% in value
   each year. Suppose the original cost is
Find the value of the car in its second year.
25% is a rate of decrease: year 2 = 75%y1
           gn = 14,800 (0.75)(2 – 1)
                 gn = 11,100
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

    Worksheet 8.1:
Formulas for sequences
  Warm-up                          5/1/08
Given explicit formula Rn = n(n + 1)
1) What is the 7th term of Rn?
2) Find R30.

3) If tn is a term in a sequence, what is the
   next term?
Go over 8.1 WS
 Finish 8.1 WS
       Calculator Tutorial
       I’m learning with you!...
  8.1 Assignment

  Section 8.1
#1-12, 13, 14, 19
    Warm-up                       5/2/08
Estimate the millionth term of each sequence to
  the nearest integer, if possible.
1) The sequence defined by an = 3n – 2
   for all positive integers n.
2) The sequence defined by b1 = 400,
   bn = 0.9n-1 for all integers > 1.
3) The sequence defined by b1 = 6, bn = 3/2bn-1
   for all integers > 1.
 CHECK 8.1
LEQ: How do you find the limit of a

     Defined as the value the function
  approaches the given value (∞,- ∞, 2, etc)

           Reading (10 minutes)
              p. 496 - 500
                 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 ∞
            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/∞ = 0

8.2 Worksheet
Warm-up                       5/5/08
1) Find the sum of the first 100
   terms of the arithmetic sequence

2) Find the sum of the first 101
   terms of this sequence.
         Interesting Facts
• Venus is the only planet that rotates
• 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.
• In Australia, Burger King is called Hungry
• 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
• The lens of the eye continues to grow
  throughout a person's life.
Check 8.2 Worksheet (HW)
      9) -5
      10) 56
      11) 32
      12) 7/4
      13) Y = 1
      14) 1
     §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.
                 Infinite Series
The number of things you add is infinite
Ex) The sum of 1(n + 1) from 0 to ∞

                 Finite Series
The number of things you add is finite
Ex) The sum of 1(n+1) 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
  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)
   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
 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
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
Use:      Sn = n/2(2a1 + (n – 1)d)
          S30 = 30/2(27500 + (30 – 1)500)
          S30 = 442,500m or 442.5 km
 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 = $25,778…a9 = $29,778
    8.3 Worksheet

      Section 8.3
     p. 507 – 508
#3 – 7, 10 – 11, 13 - 15
  Warm-up                    5/6/08
1) Find a formula for the sum Sn of
   the first n terms of the geometric
   series 1+3+9+…

2) Use the formula to find the sum
   of the first 10 terms of the series.
8.3 Assignment Answers
 3) A series is a sum of the
    terms in a sequence.
 4) A. 35 B. 31
 5) A. 77 B. 65
 6) 500,500
 7) A. $7372.50
     B. $1372.50
10)   -4
11)   873,612
13)   78
14)   19 rows, 10 left over
15)   21
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
 The sum of the finite geometric
 sequence with first term g1 and
 constant ratio r ≠ 1 is given by
         Sn = g1(1 – rn)
For finite: 0 < r < 1
*The proof for the formula can be seen
 on pg. 510 of the textbook.
        Equivalent Formula
If the rate (r) is > 1, another formula
can be used (this would be an infinite

      Sn = g1(rn – 1)
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)
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)
          Sn = 15(1 – (5/6)10)
                 1 – (5/6)
                  = 75 cm
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)

8.4 Worksheet

       Section 8.4
       p. 512 -513
#5 – 7, 10 (see Ex3), 11,
     13,14, 18 - 20
  Warm-up                 5/7/08

1) Write the first six terms of the
   geometric sequence with first
   term -2 and constant ratio 3.
2) Find the sum of the first six
   terms for #1.
          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%
• 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.
           8.4p. 512 -513
       #5 – 7, 10, 11, 13,14, 18 - 20
5) 5.98
6) 66,485.13
7) Not the million
10) 12 – 3
11)17 terms; 127.037831
18)(2i + 1) is > by 20
19)a) $25,250 b) $218.750
  b) yes; 1
Quiz over 8.1 - 8.3
  20 minutes…

 Then, read pg.
   516 - 520
Exploring Infinite Series

In class activity
     p. 515
        §8.5: Infinite Series
How do you solve problems involving infinite
 geometric series?

     What would an infinite series be?
               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 “r”< 1, the series
  converges formula: S∞ = g1
• If “r” > 1 the series diverges

8.5 Worksheet
  Warm-up                         5/8/08
1) Write the first five terms of the harmonic
2) Use a calculator to find how many terms
   of the series must be added for the sum
   to exceed 3.
3) Use a calculator to find how many terms
   of the series must be added for the sum
   to exceed 5.
4) T/F     The harmonic series is divergent.
                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.
• Stalks of sugar cane can reach up to 30 feet.
• Tasmania is said to have the cleanest air in the
• 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
 Pass back papers

Finish 8.5 Worksheet
   In Class,
   Self Test
p. 550 #1 - 11

    Chapter Review
         p. 551
# 1 – 14, 21, 22, 24 - 26,
     27 – 36, 42 - 46
  Warm-up                         5/9/08
Evaluate the arithmetic or geometric
  sequence given:
1) 103 + 120 + 137 + 154 + … + 290
2) The sum of the first 100 terms of the
   sequence (4k – 13).
3) The sum of the first 20 terms of the
   sequence 10(0.6)n – 1
               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.
• 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
• 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

• All library books are due
  by the end of today.
• Check your lockers, etc.
• Your final: 5/16 (next
Collect Chapter 8 Review
     Chapter 8 Test
 Teacher Evaluations…
          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”
             Random Facts
• Baby beavers are called kittens.
• You have no sense of smell when you're
• Ants don’t sleep.
• An albatross can sleep while it flies!
• The earth is .02 degrees hotter during a full
• By feeding hens certain dyes they can be
  made to lay eggs with multi-colored yolks.
• 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
  wasn’t a federal crime to assassinate
  the President.
• Each year, 24,000 Americans are bitten
  by rats!
Go over Chapter 8 Test

Chapter 1 “Test Form D”
   Warm-up                               5/13/08
  The following gives     Months since
                                           # www Sites

  the number of World
                               6            130
  Wide Websites
  during a period of          12            623
  years.                      18           2,738
1) Make a scatter plot.       24          10,022
2) Find a good model          30          23,500
   (linear, quadratic,        37          100,000
   etc)                       42          230,000
          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!
• 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!
 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
• 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.
• What if a senior is not exempt from the
  exam and is absent the day of the
  –If a senior is absent for a Friday
    exam, he’ll have to make it up on
    Monday (a second version of the
  –If a senior is absent for both days of
    senior exams, they’ll have to take the
    exams on Tuesday and Wednesday
    with the underclassmen.
• 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
ALL grades for ALL seniors
should be in Power Grade
no later than 12:00 noon on
Monday (with the few
exceptions resulting from #3
or #4 above).
Seniors have limited activities next
week. Only seniors who are taking
final exams should be in the
building (i.e., it’s not time for them
to hang out in your class because
they’re “done” with high
school). After each activity, seniors
will be excused from school for the
remainder of the day.
• Monday, May 19 – Fun photo day, 9:00
  AM. Some group shots will be taken in
  and possibly around the stadium.
• Tuesday, May 20 – Graduation Practice,
  8:30 AM, Roquemore Field
• Wednesday, May 21 – Graduation
  Practice, 8:30 AM, Roquemore Field
• Thursday, May 22 – Senior Breakfast
  (optional, RSVP to senior homeroom
  teacher by Monday, May 19).
         Plan for Algebra III
• “Project Folders”
• Four Projects Total
• Turned in by your last day (if you’re a
  senior & exempting my exam, that will be
• If you’re a senior and taking the final
  exam, I will give you a study guide
   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

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