M1410901and06

M1410901and06 - (Chapter 9 Discrete Math 9.01 CHAPTER 9...

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(Chapter 9: Discrete Math) 9.01 CHAPTER 9: DISCRETE MATH Calculus tends to deal more with “continuous” mathematics than “discrete” mathematics. What is the difference? Analogies may help the most. Discrete is digital; continuous is analog. Discrete is a dripping faucet; continuous is running water. Discrete math tends to deal with things that you can “list,” even if the list is infinitely long. Math 245 is the Discrete Math course at Mesa. SECTION 9.1: SEQUENCES AND SERIES, and SECTION 9.6: COUNTING PRINCIPLES PART A: SEQUENCES An infinite sequence can usually be thought of as an ordered list of real numbers. It is often denoted by either: a 1 , a 2 , a 3 , or a 0 , a 1 , a 2 , (this is more common in Calculus) where each of the “ a i ”s represent real numbers called terms. If we’re lucky, there is a nice pattern to our sequence of numbers – a pattern that we can describe using simple mathematical expressions.

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(Chapter 9: Discrete Math) 9.02 Example Write the first three terms of the sequence with general n th term a n = n 3 1 . Assume that we begin with n = 1 . Solution n = 1 : a 1 = 1 ( ) 3 1 = 0 n = 2 : a 2 = 2 ( ) 3 1 = 7 n = 3 : a 3 = 3 ( ) 3 1 = 26 The first three terms are: 0, 7, and 26. We may graph terms as the points n , a n ( ) , just as we used to graph points x , f x ( ) ( ) . Technical Note: Observe that the graph of the sequence is a “discretized” version of the graph of f x ( ) = x 3 1 , where the domain is the set of positive integers. In this sense, sequences may be thought of as functions.
(Chapter 9: Discrete Math) 9.03 PART B: SIGN ALTERNATORS Alternating sequences have terms that alternate (i.e., consistently switch) between positive and negative terms. They arise frequently in Calculus. Example Write the first four terms of the sequence where a n = 1 ( ) n n . Assume that we begin with n = 1 . Solution n = 1 : a 1 = 1 ( ) 1 1 = 1 n = 2 : a 2 = 1 ( ) 2 2 = 2 n = 3 : a 3 = 1 ( ) 3 3 = 3 n = 4 : a 4 = 1 ( ) 4 4 = 4 Recall that the parity of an integer is either “even” or “odd.” The parity of n here determines the sign of the term. Example How can we obtain an alternating sequence like the one above, but starting with a positive term? Let’s adjust our sign alternator. Try: a n = 1 ( ) n + 1 n , or a n = 1 ( ) n 1 n In either case, we obtain: 1, 2 , 3, 4 , … PART C: EVEN VS. ODD 2 n yields even integers, where n Z .

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M1410901and06 - (Chapter 9 Discrete Math 9.01 CHAPTER 9...

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