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SECTION 7.2 ARITHMETIC SEQUENCES

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(a) 5, 9, 13, 17, 21,25 (b) 2, 2.5, 3, 3.5, 4, 4, 4.5 4.5 (c) 8, 5, 2, - 1, - 4, - 7 Adding 4 Adding.5 Adding - 3 Arithmetic Sequences have a “common difference”. (a) 4(b).5 (c) - 3

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ARITHMETIC SEQUENCE RECURSION FORMULA a n = a n - 1 + d This formula relates each term in the sequence to the previous term in the sequence. a n = a n - 1 + 4 b n = b n - 1 +.5c n = c n - 1 - 3

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EXAMPLE: Given that e 1 = 4 and the recursion formula e n = e n - 1 + 0.3, determine the first five terms of the sequence { e n }. e 1 = 4 e 2 = 4 +.3 = 4.3 e 3 = 4.3 +.3 = 4.6 e 4 = 4.6 +.3 = 4.9 e 5 = 4.9 +.3 = 5.2

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Recursion Formulas have a big disadvantage. In the last example, what would happen if we needed to know the value of the 291st term? Explicit Formulas are much better for finding nth terms.

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ARITHMETIC SEQUENCE EXPLICIT FORMULA a 1a 1a 1a 1 a1a1a1a1 a 2a 2a 2a 2 a1a1a1a1 d a 3a 3a 3a 3 a1a1a1a1 d d a 4a 4a 4a 4 a1a1a1a1 d d d a na na na n a1a1a1a1 d 1 d 2 d’s 3 d’s n-1 d’s n-1 d’s

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a n = a 1 + (n - 1) d Example:Determine e 291 for the arithmetic sequence with e 1 = 4 and common difference d = 0.3 e 291 = 4 + (291 - 1) (0.3) 91 ARITHMETIC SEQUENCE EXPLICIT FORMULA

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SUMS OF ARITHMETIC SEQUENCES 1 + 2 +... + 49 + 50 + 51 + 52 +... + 99 + 100 50 PAIRS OF 101 50(101) = 5050

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ARITHMETIC SEQUENCE SUM FORMULA The sum of n terms of an arithmetic sequence is n times the average of the first and last terms to be added.

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EXAMPLE: Determine the sum of the first 200 terms of the arithmetic sequence { a n } with a 1 = - 5 and d = 3. First, we must find a 200 a 200 = - 5 + (199)(3) = 592

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EXAMPLE: a 1 = - 5 and a 200 = 592 58, 700

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FINDING THE NUMBER OF TERMS IN A SEQUENCE 4, 9, 14, 19,..., 64 Just add 5 on the calculator until you get to 64 and see how many terms there are in the sequence. OR…

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FINDING THE NUMBER OF A TERMS IN A SEQUENCE

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EXAMPLE (2 - 1) + (4 - 1) + (6 - 1) +... + (100 - 1) 1 + 3 + 5 +... + 99 1 + 3 + 5 +... + 99 This is the sum of an arithmetic sequence

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EXAMPLE 1 + 3 + 5 +... + 99

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CONCLUSION OF SECTION 7.2 CONCLUSION OF SECTION 7.2

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