M1410902and03

M1410902and03 - (Chapter 9 Discrete Math 9.11 SECTION 9.2...

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(Chapter 9: Discrete Math) 9.11 SECTION 9.2: ARITHMETIC SEQUENCES and PARTIAL SUMS PART A: WHAT IS AN ARITHMETIC SEQUENCE? The following appears to be an example of an arithmetic (stress on the “me”) sequence: a 1 = 2 a 2 = 5 a 3 = 8 a 4 = 11 We begin with 2. After that, we successively add 3 to obtain the other terms of the sequence. An arithmetic sequence is determined by: • Its initial term Here, it is a 1 , although, in other examples, it could be a 0 or something else. Here, a 1 = 2 . • Its common difference This is denoted by d . It is the number that is always added to a previous term to obtain the following term. Here, d = 3 . Observe that: d = a 2 a 1 = a 3 a 2 = = a k + 1 a k k Z + ( ) = The following information completely determines our sequence: The sequence is arithmetic. (Initial term) a 1 = 2 (Common difference) d = 3
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(Chapter 9: Discrete Math) 9.12 In general, a recursive definition for an arithmetic sequence that begins with a 1 may be given by: a 1 given a k + 1 = a k + d k 1; " k is an integer" is implied ( ) Example The arithmetic sequence 25, 20, 15, 10, … can be described by: a 1 = 25 d = 5
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(Chapter 9: Discrete Math) 9.13 PART B: FORMULA FOR THE GENERAL n th TERM OF AN ARITHMETIC SEQUENCE Let’s begin with a 1 and keep adding d until we obtain an expression for a n , where n Z + . a 1 = a 1 a 2 = a 1 + d a 3 = a 1 + 2 d a 4 = a 1 + 3 d a n = a 1 + n 1 ( ) d The general n th term of an arithmetic sequence with initial term a 1 and common difference d is given by: a n = a 1 + n 1 ( ) d Think: We take n 1 steps of size d to get from a 1 to a n . Note: Observe that the expression for a n is linear in n . This reflects the fact that arithmetic sequences often arise from linear models. Example Find the 100 th term of the arithmetic sequence: 2, 5, 8, 11, … (Assume that 2 is the “first term.”) Solution a n = a 1 + n 1 ( ) d a 100 = 2 + 100 1 ( ) 3 ( ) = 2 + 99 ( ) 3 ( ) = 299
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(Chapter 9: Discrete Math) 9.14 PART C: FORMULA FOR THE n th PARTIAL SUM OF AN ARITHMETIC SEQUENCE The n th partial sum of an arithmetic sequence with initial term a 1 and common difference d is given by: S n = n a 1 + a n 2 Think: The (cumulative) sum of the first n terms of an arithmetic sequence is given by the number of terms involved times the average of the first and last terms. Example Find the 100 th partial sum of the arithmetic sequence: 2, 5, 8, 11, … Solution We found in the previous Example that: a 100 = 299 S n = n a 1 + a n 2 S 100 = 100 ( ) 2 + 299 2 = 100 ( ) 301 2 = 15,050 i.e., 2 + 5 + 8 + ... + 299 = 15,050 This is much easier than doing things brute force on your calculator!
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M1410902and03 - (Chapter 9 Discrete Math 9.11 SECTION 9.2...

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