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)
Disclaimer…
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?
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
Concept II
Arithmetic Series
Lesson Essential Question (LEQ):
How do you solve problems involving
arithmetic series?
Vocabulary:
Infinite series, finite series, arithmetic
series,
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,
Concept IV
Combinations
Lesson Essential Question (LEQ):
How do you use series to find
combinations?
Vocabulary:
combination
Concept V
Pascal’s Triangle and Binomial
Theorem
Lesson Essential Question (LEQ):
How is Pascal’s Triangle used to
expand polynomials?
Vocabulary:
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
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(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?
16,19,22,…127
a1 = 16
d=3
an = 127
127 = 16 + (n – 1)3
127 = 3n + 13
N = 38
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
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
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 = 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
Homework
Worksheet 8.1:
Formulas for sequences
#1-6
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
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http://education.ti.com/educationpo
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onlinealgebra_free.html?bid=2
8.1 Assignment
Section 8.1
P.493
#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
n+1
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
ASSIGNMENT
§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 (∞,- ∞, 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
Assignment
8.2 Worksheet
Warm-up
5/5/08
1) Find the sum of the first 100
terms of the arithmetic sequence
3,7,11,…
2) Find the sum of the first 101
terms of this sequence.
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.
• 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.
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
sequence.
Ex1)
Ex2)
Ex3)
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)
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
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
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
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 = $25,778…a9 = $29,778
Practice
8.3 Worksheet
Homework:
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)
11)
13)
14)
15)
-4
873,612
78
19 rows, 10 left over
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
sequence.
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 < 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
series).
Sn = g1(rn – 1)
r-1
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)
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
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
Practice
8.4 Worksheet
Assignment
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,-6,-18,-54,-162,-486
2) Find the sum of the first six
terms for #1.
-728
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.
• 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
13)4,265.625
13)-33.25
18)(2i + 1) is > by 20
19)a) $25,250 b) $218.750
20)2,4/3,8/7,16/15,32/31
b) yes; 1
Questions?
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?
Recall:
Divergent
Convergent
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
1–r
• If “r” > 1 the series diverges
Practice
8.5 Worksheet
Warm-up
5/8/08
1) Write the first five terms of the harmonic
series.
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
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.
Pass back papers
Finish 8.5 Worksheet
In Class,
Complete
Self Test
p. 550 #1 - 11
HW
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
2358
2) The sum of the first 100 terms of the
sequence (4k – 13).
18,900
3) The sum of the first 20 terms of the
sequence 10(0.6)n – 1
24.999
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
it.
• 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 nonsmokers 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.
Reminders:
• All library books are due
by the end of today.
• Check your lockers, etc.
• Your final: 5/16 (next
Friday)
Questions?
Collect Chapter 8 Review
(E.C.)
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
sleeping!
• Ants don’t 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.
• 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
The following gives
the number of World
Wide Websites
during a period of
years.
1) Make a scatter plot.
2) Find a good model
(linear, quadratic,
etc)
5/13/08
Months since
1/1/1993
# www Sites
6
12
18
24
30
37
42
130
623
2,738
10,022
23,500
100,000
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
Friday.
• 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
exam?
–If a senior is absent for a Friday
exam, he’ll have to make it up on
Monday (a second version of the
exam).
–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
retake.
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).
NO PARTIES and NO
“FREE” DAYS.
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
tomorrow!)
• If you’re a senior and taking the final
exam, I will give you a study guide
tomorrow
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