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Understanding Sequences, Series & Counting Methods

Learn about finite and infinite sequences, arithmetic progressions, series convergence, and more with examples and practical applications including Gauss's insights. Solve counting problems to enhance your mathematical skills.

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Understanding Sequences, Series & Counting Methods

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  1. Block 3 Discrete Systems Lesson 7a – More Counting Sequences, series and much more one two three four five six seven eight nine ten

  2. Defining Sequences • Sn is the nth term in a sequence that may be finite or infinite • Sn is a function defined on the set of natural numbers • Examples: • If Sn = 1/n, then the first 4 terms in the sequence are 1, 1/2, 1/3, 1/4 • If Sn = 1/n!, then the first 4 terms in the sequence are 1, 1/2, 1/6, 1/24 • The general term for the sequence -1, 4, -9, 16, -25 is Sn = (-1)n n2

  3. Arithmetic Progression • Sn = a + (n-1)(d) is an arithmetic progression starting at a and incrementing by d • For example, the first six terms of Sn = 3 + (n-1) (4) are 3, 7, 11, 15, 19, 23

  4. The limit of an infinite sequence If for an infinite sequence, s1, s2, …, sn, … there exists an arbitrarily small  > 0 and an m > 0 such that |sn – s| <  for all n > m, then s is the limit of the sequence.

  5. Examples

  6. Series • The sum of a sequence is called a series. • The sum of an infinite sequence is called an infinite series • If the infinite series has a finite sum, then the series is said to converge; otherwise it diverges Let Sn = s1 + s2 + … + sn Sn is the sequence of partial sums

  7. The Geometric Series Sn = a +ar + ar2 + … + arn-1 r Sn = ar + ar2 + … + arn + arn Sn - r Sn =(1-r) Sn = a - arn This is a most important series.

  8. The Geometric Series in Action • Find the sum of the following series:

  9. Carl Friedrich Gauss (April 30, 1777 – February 23, 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. Sometimes known as "the prince of mathematicians", Gauss had a remarkable influence in many fields of mathematics and science and is ranked beside Euler, Newton and Archimedes as one of history's greatest mathematicians.

  10. More about Carl In elementary school Gauss’ teacher tried to occupy pupils by making them add up the integers from 1 to 100. The young Gauss produced the correct answer within seconds by a flash of mathematical insight, to the astonishment of all. Gauss had realized that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050. While the story is mostly true, the problem assigned by Gauss's teacher was actually a more difficult one.

  11. The Arithmetic Series The sum of the first 100 odd numbers is

  12. Other series worth knowing Binomial series, n a positive integer exponential series logarithmic series

  13. 16 great problem exercises in Counting. Yes, 16 - just count them! “To be able to count is a necessary first step in becoming an engineer manager.”- Ima Counter, circa 74 AD According to the International Programs Center, U.S. Bureau of the Census, the total population count of the World on 02/04/06 at 19:54 GMT (EST+5) was projected to be 6,495,680,642

  14. Problem Exercise #1 • How many ways can a judge award first, second, and third places in a contest with 20 contestants? (20)(19)(18) = 6,840

  15. Problem Exercise #2 • The ENM 645 course final exam requires that the student answer 10 out of 13 questions. • How many choices are there? • How many choices if the first 2 questions must be answered? • How many if the 1st or 2nd must be answered but not both? • How many if exactly 3 of the first 5 questions must be answered? • How many if at least 3 of the first 5 questions must be answered? (i) C(13,10) = 286; (ii) C(11,8) = 165 (iii) 2 C(11,9) = 110 (iv) C(5,3) C(8,7) = 80 (v) C(5,3)C(8,7) + C(5,4)C(8,6) + C(5,5)C(8,5) = 276

  16. Problem Exercise #3 • How many 6-character computer passwords can be formed where the first character must be a letter and the remaining characters may a letter or a digit but no special characters where • all are lower case letters? • upper and lower case are distinguishable? (i) (26) (36)5 = 1,572,120,576 (ii) (52) (62)5 = 47,638,907,264

  17. Problem Exercise #4 • Mr. Hi N. Mitey, the Chief Engineer for the Dell Phye Company, plans to visit 3 of the 7 technical centers this week. • In how many ways can he plan his itinerary? • In how many can he choose the sites to visit? • In how many can he choose the sites to visit if he plans to visit at least one of the 2 sites that are in trouble? (i) (7)(6)(5) = P(7,3) = 7! / 4! = 210 (ii) C(7,3) = 7! / [4! 3!] = 210/6 = 35 (iii) 2 C(5,2) + C(5,1) = 20 + 5 = 25 Mr. Hi N. Mitey

  18. Problem Exercise #5 • A team is being formed to perform a Failure Mode and Effects Analysis (FMEA) consisting of 3 design engineers, a reliability engineer, 2 quality control specialists, a production supervisor, and a manager. How many ways can the team be formed from a pool of 10 design engineers, 2 reliability engineers, 3 quality control specialists, 4 production supervisors and 3 managers? C(10,3) C(2,1) C(3,2) C(4,1) C(3,1) = (120)(2)(3)(4)(3) = 8,640 The FMEA Team

  19. Problem Exercise #6 • MacFadden’s, a fast food hamburger chain, advertises that their BigFad, a full quarter-pounder can be prepared in 256 different ways. How can that be? all beef patty special sauce, lettuce, cheese, pickles, onions seasoning on a sesame seed bun. 28 = 256

  20. Problem Exercise #7 • A reliability test consists of subjecting 3 prototype powertrains to 4 different temperatures, 3 humidity levels, and 2 vibration settings. In addition, each test will also be run with and without contaminates present. How many tests must be performed? 3 x 4 x 3 x 2 x 2 = 144

  21. Problem Exercise #8 • How many ways may five tires be rotated on a car? • How many ways may 4 tires from five tires be rotated on a car? • How many ways can 4 tires be selected from 5 to be put on a car? This one goes on the right front. • P(5,5) = 120 • (ii) P(5,4) = 120 • (iii) C(5,4) = 5

  22. Problem Exercise #9 • How many 5-card poker hands are there? • How many contain exactly one pair? I am not a betting man. The odds are not in my favor. • C(52,5) = 2,598,960 • (ii) C(13, 4)C(4, 1)C(4, 2)C(4, 1)3 [xx y z w] • = 715*4*6*64 = 1,098,240 • C(13,4) ~ types chosen from 13 in deck • C(4, 1) = which type is pair • C(4, 2) = way to chose 2 of the 4 which is the pair • C(4,1)3 = ways to chose one each of remaining 3 types.

  23. Problem Exercise #10 • If there are 10 horses running at Aqueduct race track, how many selections are there if you bet on a • quinella? • exacta? • trifecta? (i) 2 horses in either order = C(10,2) = 45 (ii) 2 horses exact order = (10) (9) = 90 (iii) 3 horses exact order: (10)(9)(8) = 720

  24. Problem Exercise #11 – a look ahead • Harry and Sally have been invited to a Valentine’s Day party. They both are to be seated at a round table with 6 other guests. Harry would like to sit next to Sally. How likely is that if the seating is randomly assigned? I hope we can sit together at the party. 7! Ways to seat 8 people at round table 2 x 6! Ways for Harry & Sally to sit together Prob = 2 (6!) / 7! = 2/7 = .2857

  25. Problem Exercise #11a • But what happens if they are to be seated on one side of a long table for 8? I heard they are seating us at the head table now. 8! Ways to seat 8 people at head table 2 x 7 x 6! Ways for Harry & Sally to sit together _ _ _ _ _ _ _ _ Prob = 2 (7) (6!) / 8! = 2/8 = .25

  26. Problem Exercise #12 a point • There are 12 points, A, B, …, in a plane, no 3 on the same line • how many lines are determined by the points? C(12,2) = 66 • how many lines pass through the point A? 11 lines • how many triangles are determined by the points? C(12,3) = 220

  27. Problem Exercise #13 • A job shop receives 5 jobs each to be processed on 4 machines. If each job can be processed on the 4 machines in any order, how many different job schedules are possible? This is an NP-hard problem. (5!)4 = 207,360,000 I need that job schedule in an hour!

  28. Problem Exercise #14 • In a five card poker hand, how many ways can a full house be dealt? No one will beat this full house! [xxx yy] C(13, 2)C(2, 1)(4,3)C(4,2) = 3,744 C(13, 2) ~ 2 types in hand) C(2, 1) ~ which type is 3 C(4, 3) ~ choosing 3 of 4 C(4, 2) ~ choosing 2 of 4

  29. The Odds - 5 cards randomly drawn from a full deck of 52

  30. Problem Exercise #15 • How many subsets can be formed from the following set? A = {a, b, c, d, e, f, g, h}? • Power Set 28 = 256 A set of Allen wrenches

  31. Problem Exercise #16 • Professor Notso Brite tells 3 jokes every year in his statistics class. If his policy is to never tell the same 3 jokes in any year, what is the minimum number of jokes he will need over his 30 year teaching career? Have you heard the one where a doctor, teacher, and lawyer go into a bar; and… C(n, 3) ≥ 30 Find n! C(6,3) = 20 C(7,3) = 35 Answer: n = 7

  32. Our Counting has come to an end. Remember, I am counting on you! Let me count the ways … -olde English saying, anonymous, circa1600 How do I love thee, let me count the ways, C(100, 50), 100! Googolgoogol … Permutations of permutations

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