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Lesson 10.2: Arithmetic Sequences & Series Sequences –Algebraic formulaspecific terms –Recursion or Recursive formulas –Arithmetic means Series –Sum of finite arithmetic sequences

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1. 5, 9, 13, 17, _____ 2. 2, -5, -12, -19, _____ Arithmetic Sequence: a sequence whose consecutive terms have a common difference, d. d = 4 d = For 1: a 1 = 5, d = 4, so a n = 5 + (n – 1)4 = For 2: a 1 = 2, d = -7, so a n = = a 1 + dn – d = dn + (a 1 – d) The formula for a n can be written given a 1 and d: a n = a 1 + (n – 1)d a n = dn + c (where c = a 1 – d) simplified form

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3. a 3 = 63, d = 4 Write the explicit formula for a n a n = a 1 + (n – 1)d a n = dn + c (where c = a 1 – d) p675 #30. a 5 = 190, a 10 = = 4*3 + c 51 = c a 3 = dn + c so, a n = 4n + 51 a 10 = a 5 + (10 – 5)d 115 = d -75 = 5d -15 = d a 5 = dn + c 190 = -15*5 + c 265 = c so, a n = -15n find d

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Write the first 5 terms of the arithmetic sequence. (need the formula or a 1 & d ) a n = a 1 + (n – 1)d a n = dn + c (where c = a 1 – d) p675 #39. a 8 = 26, a 12 = a n = 4n + 51 a 12 = a 8 + (12 – 8)d a 8 = a 1 + (n – 1)d a 1 = -2 a 2 = a 1 + d = = 2 a 3 = a 2 + d = a 4 = a 3 + d = a 5 = a 4 + d = Have formula, so use it! a 1 = 4(1) + 51 = 55 a 2 = 4(2) + 51 = 59 etc. No formula, so find a 1 & d Sequence: find d find a 1

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1. 5, 9, 13, 17, _____ 2. 2, -5, -12, -19, _____ A recursion formula defines a n in terms of previous term(s). d = 4 d = -7 For arithmetic sequences, a n = a n-1 + d (must also provide a term in the sequence). 3. a 3 = 63, d = 4 p674 #30. a 5 = 190, a 10 = 115 d = -15 a 1 = 5, a n = a n-1 + 4

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Arithmetic means are terms between any nonconsecutive terms in an arithmetic sequence. 1. Find the 4 arithmetic means between 72 and ,,,,, 52 a1a1 a6a6 a n = a 1 + (n – 1)d a n = dn + c (where c = a 1 – d) a 6 = a 1 + (n – 1)d 52 = 72 + (5)d -20 = 5d -4 = d Need d. So, 72, 68, 64, 60, 56, Find the 3 arithmetic means between 21 and 45. d = 6, so 21, 27, 33, 39, 45

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Arithmetic series: the sum of a finite arithmetic sequence. The sum of a finite arithmetic sequence is or See derivation on p670.note: a n = a 1 + (n – 1)d Examples: 1. Find the sum of the 1 st 20 even integers beginning at Find Have n =, a 1 =, d =, so use 2 nd formula. Formula looks like an arithmetic series, expand a few terms to be sure: 2*3 + 1 = 7, 2*4 + 1 = 9, 2*5 + 1 = 11, … Can easily determine n, a 1, & a n, so use 1 st formula. n = a 1 = [k = 3] a 108 = [k = 110] be careful!

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Homework: Arithmetic Sequences: pp # 2, 5, 7, 20, 24, 34, 38, 62, 64 Arithmetic Series: p675 #42, 44, 48, 50, 54, 66, 70, 72

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