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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 I’m learning with you!... http://education.ti.com/educationpo rtal/sites/US/nonProductMulti/pd_ 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) -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 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 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. 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 5/13/08 The following gives Months since 1/1/1993 # 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 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