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Warm-up 4/30/08

Warm-up 4/30/08. Write the first six terms of the sequence with the given formula. a 1 = 2 a n = a n – 1 + 2n – 1 2) a n = n 2 + 1 3) What do you notice about your answers to Questions 1 and 2?. Copy SLM for Unit 7 (chapter 4, 5) Disclaimer…. §8.1: Formulas for sequences.

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Warm-up 4/30/08

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  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. §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)

  4. 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?

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. Homework Worksheet 8.1: Formulas for sequences # 1 - 6

  12. 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?

  13. Go over 8.1 WS Finish 8.1 WS

  14. Calculator Tutorial I’m learning with you!... http://education.ti.com/educationportal/sites/US/nonProductMulti/pd_onlinealgebra_free.html?bid=2

  15. 8.1 Assignment Section 8.1 P.493 #1-12, 13, 14, 19

  16. 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 > 1. 3) The sequence defined by b1 = 6, bn = 3/2bn-1 for all integers > 1.

  17. CHECK 8.1 ASSIGNMENT

  18. §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

  19. 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

  20. Assignment 8.2 Worksheet

  21. 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.

  22. 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.

  23. 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.

  24. Check 8.2 Worksheet (HW) • -5 • 56 • 32 • 7/4 • Y = 1 • 1

  25. §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.

  26. 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.

  27. Ex1)

  28. Ex2)

  29. Ex3)

  30. 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)

  31. 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

  32. 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

  33. 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

  34. 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

  35. Practice 8.3 Worksheet Homework: Section 8.3 p. 507 – 508 #3 – 7, 10 – 11, 13 - 15

  36. Warm-up 5/6/08 • Find a formula for the sum Sn of the first n terms of the geometric series 1+3+9+… • Use the formula to find the sum of the first 10 terms of the series.

  37. 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

  38. -4 • 873,612 • 78 • 19 rows, 10 left over • 21

  39. 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.

  40. 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.

  41. 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

  42. 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)

  43. 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

  44. 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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