Do in calculator, too.

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3 You do Do in calculator, too.

4 Sequence formulas that give you the exact definition of a term are explicit formulas. Formula: a n = 5n Explicit, every term is defined by this formula. Recursive formulas define terms based on preceding terms. Formula: a a 1 2 = 1 = 1 a = a + a, for_ n n n 1 n 2 3 Recursive, after the second term and the two previous terms to get the next term.

5 Formula: a a 1 2 = 1 = 1 a = a + a, for_ n n n 1 n 2 3 Find the first 6 terms of the sequence. a a 1 2 = 1 = 1 a = a + a = 1+ 1 = a = a + a = = a = a + a = 3+ 2 = a = a + a = 5+ 3 =

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7 You do..

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9 Problems for you.. Page ,5,9,13,17,21,25,27,31,35,37, 45,51,53

10 More problems to do. Page ,59,61,63,65,67,69,71,73,77

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12 Notation for multiple sums: sigma Definition_ of _ Sigma_ Notation: The_ sum_ of _ n_ terms_ a, a, a,..., a is_ written_ as n ai = a1 + a2 + a a i= 1 i = index_ of _ summation a = ith_ term_ of _ the_ sum i lower_ bound = 1 upper_ bound = n n n

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14 Summation Formulas n i= 1 c = cn 3 5 = = 5( 3) = 15 i= 1 n i= 1 i 2 = n n n + i = ( 1) 2 i= 1 n( n + 1)( 2n + 1) 6 4! i 2 = 4(5)(9) = 30 " = 30 6 i=1 ( ) 3 3(3 +1)! i = = 6 " = 6 2 i=1 ( ) n i= 1 i 3 = n 2 ( n + 1) i 3 = 22 (2 +1) 2 i= ( )! = 9 " = 9

15 Evaluate_ the_ sum n i i= 1 n for_ n = 10_ & _ n = 1000 n i +1! = 1 n!( i +1) " by _Theorem n 2 i=1 n 2 i=1 n 1!( i +1) = 1 n n " %! i +! 1 n 2 i=1 n 2 $ ' # i=1 i=1 & ( by _Theorem 1 n 2 n n " % $! i +! 1' # i=1 i=1 & = 1 " n(n +1) n 2 # $ n % + n & ' ( by _Theorem

16 Evaluate_ the_ sum n i i= 1 n for_ n = 10_ & _ n = n 2 n n " % $! i +! 1' # i=1 i=1 & = 1 " n(n +1) n 2 # $ n % + n & ' ( by _Theorem 1! n(n +1) $ + n n 2 " # n % & = n + 3 2n when _ n = 10 ' =.65 when _ n = 1000 ' =.50015

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19 Problems for you. Page odds

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27 Problems for you From page 598 #1-9 odds, 13, 15, 19, 23, 27, 29, 31, 35, 37

28 Problems for you. Page odds

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32 You do.

33 You do.

34 Calculator List Math 5 List Ops 5

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36 You do. (by hand and calculator) Problems for you. Page 607 1,5,9,13,17,19,23,27,31,33,35

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38 We need to do a cumulative sum for an infinite geometric Cumsum under LIST OPS Scroll right

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40 An example of inductive reasoning Looking for patterns

41 Some of these patterns hold for several early terms but break down afterwards. One of the most famous of these is Fermat s conjecture. These are all primes. One of the troubles with evaluating Fermat s conjecture was that the numbers grow very large very quickly

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47 You do.

48 Problems for you.. Pg all

49 Part 2

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51 You do.

52 Let s do the first one together.

53 Showing it works for when 1 i= 1 i = 1 n(n + 1) = 2 (1)(2) 2 n = 1: = 1 check. Showing if it works for nth case it works for n + 1case : Formula for n + 1case : (n + 1)((n + 1) + 1) 2 ( n + 1)( n + 2) = 2 n + i= n + ( n + 1) = n( n + 1) + ( n + 1) 2 = n( n + 1) 2( n + 1) = n 2 + n + 2n + 2 = 2 n 2 + 3n = n 2 + 3n = ( n + 1)( n + 2) 2 check

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55 If you are using Mathematical induction to show that something holds for n greater than some number, say for 4, then.. 1. Show it holds for the case when n = 4 2. Show if it holds for the case n then that it holds for the case n+1..(exactly like before)

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57 Problems for you.. From pg odds

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62 Note:

63 You do = = = = = ( ) ) (1)(2 ) (2 (4) ) (2 (6) ) (2 (4) (1) = + x x x x x

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65 You do.

66 Problems for you.. Page odds, odds

67 Part 2 Finding Binomial Coefficients in the Calculator

68 An example.

69 One more example You do

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72 Problems for you. Page all (use calculator) odds

73 Problem: With a simple counting problem, it usually easiest to list the possible ways the result can be found. (3,3), (2,4), (4,2), (1,5), (5,1): 5 ways distinct means different, no doubles of the same number (1,9), (9,1), (2,8), (8,2), (3,7), (7,3), (4,6), (6,4): 8 ways

74 You try For a problem like this it would be unwise to count all the possibilities. Use Reason! (30 possibilities for 1 st number) x (30 possibilities for 2 nd number) x (30 possibilities for 3 rd number) 27,000 possibilities!

75 How many different local telephone numbers are possible (disregard the area code)? Note: The first two numbers of a local number cannot be 0 or 1.

76 Permutations are the number of ways to order n elements, an important subset of the Fundamental Counting Principle

77 You do. Remember, with Permutations order matters!

78 Last problem reworked using the above formula

79 Distinguishable permutations mean they are unique Question: How many ways can you arrange the letters A,B,C,D? All permutations are different. 4 x 3 x 2 x 1 = 24 ways A as 1 st letter: B as 1 st letter: C as 1 st letter: D as 1 st letter: ABCD BACD CBAD DBCA ABDC BADC CBDA DBAC ACDB BCDA CADB DCAB ACBD BCAD CABD DCBA ADBC BDAC CDBA DABC ADCB BDCA CDAB DACB All these arrangements are unique: no repeats

80 Distinguishable permutations mean they are unique Question: How many ways can you arrange the letters A,A,B,D? AABD where the first A is used and AABD where the second A is used are NOT Distinguishable! AABD BAAD DAAB ABAD BADA DABA ABDA BDAA DBAA AADB ADAB ADBA 12 distinguishable ways or 24 undistinguishable ways

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82 You do. Page odds

83 Part II Recall that with Permutations, the order of the chosen items was crucial When you choose a subset of items from a set, where order does not matter, then you want Combinations. 10 ways

84 AB, AC, AD, AE, AF, AG, BC, BD, BE, BF, BG, CD, CE, CF, CG, DE, DF, DG, EF, EG, FG: 21 ways Note: this is the same formula as the binomial coefficient

85 How many 5-card poker hands are there? There are 52 cards and you choose 5 (order does not matter) You do. For both problems order does not matter. You should start thinking of that first (Does order matter?) when you look at counting problems.

86 Choosing seven boys

87 You do. Page odds

88 Let s start by learning the terms. An experiment is something that happens whose result is uncertain. The set of all results or outcomes is called the Sample Space of an experiment. Any subset of the Sample Space is called an Event.

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90 You do. The Probability of an event (where all the outcomes are equally likely) equals the ratio of the outcomes of the event to all the possible outcomes. ½ because there are two ways for this to happen (TH or HT) out of a total of Sample Space of 4 items (TT,HH,TH,HT), so 2/4.

91 ¼ because there are 13 hearts in a deck of 52 total cards and 13/52 equals ¼.

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93 Use the Fundamental Counting Principle to Determine the Sample Space Size 6 x 6 = 36 Figure how many ways to make 7

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97 More Probability Choosing an item at random, means all items are equally likely to be selected. Since the graduate is chosen at random, each graduate is equally likely to be selected.

98 Number of Colleges in Various Regions

99 Mutually Exclusive Events have no outcomes in common. Rolling a 6 on a die and choosing a spade from a deck of cards are mutually exclusive. Rolling a 6 on one die and having the sum of the two dice you rolled equally 4 are not mutually exclusive events.

100 Mutually exclusive? No, a card can be both a face card and a heart.

101 You do. Mutually exclusive because an employee cannot belong to more than one row. You cannot have both 0-4 and 5-9 years of service.

102 You do.

103 Problems for you.. Page #31-39 odds

104 Probability Part 3 If the computer is really generating random numbers, then each number should be chosen independent of the others.

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106 The complement of event A, labeled A, is the collection of all outcomes in the sample space that are NOT in A.

107 Sometimes it is easier to find the probability of the complement than the probability of the event you are interested in. The complement would be that none of the units produced are faulty. Probability that the unit is produced correctly. Probability that all 200 are produced correctly.

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109 Problems for you Pages #41-53 odds