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Arithmetic Sequences Notes

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Definitions

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Sequence is a set of numbers arranged in a particular order. Term is one of the set of numbers in a sequence. Arithmetic Sequence is a sequence in which the difference between two consecutive numbers is constant. Common difference is this constant difference. Explicit Formula for an Arithmetic Sequence: a n = a 1 + (n – 1)d, where a 1 is the 1 st term of the sequence, n is the # of the term, and d is the common difference. Recursive Formula for an Arithmetic Sequence: a 1 = #, a n = a n − 1 + d

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1.a 1 = 4 and the common difference is 2. a.Write the recursive formula. a 1 = 4, a n = a n − 1 + 2 b.Find the 1 st five terms of the sequence. 4, 6, 8, 10, 12 c.Write the explicit formula. a n = 4 + (n – 1)(2) Example 1

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d.Find the ninth term. a 9 = 4 + (9 – 1)(2) a 9 = 4 + 16 a 9 = 20

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a.Write the recursive formula. a 1 = 5, a n = a n − 1 – 1 b.Find the 1 st five terms of the sequence. 5, 4, 3, 2, 1 c.Write the explicit formula. a n = 5 + (n – 1)(−1) d.Find a 7. a 7 = 5 + (7 – 1)(−1) a 7 = −1

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a.Find the explicit formula. a n = −5 + (n – 1)(−5) b.Find a 16. a 16 = −5 + (16 – 1)(−5) a 16 = −80

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Numbers m 1, m 2, m 3, … are called arithmetic means between a and b if a, m 1, m 2, m 3,…, b forms an arithmetic sequence.

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1. Find the arithmetic mean between 10 and 50. Example 2

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2. Find the four arithmetic means between 35 and 70. 35, m 1, m 2, m 3, m 4, 70 is the arithmetic sequence. a 1 = 35; n = 6; a 6 = 70 70 = 35 + (6 – 1)d 35 = 5d d = 7 35, 42, 49, 56, 63, 70

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Find a 1 and d for the arithmetic sequence with the given terms. 1.a 3 = 10 and a 6 = 19 19 = a 1 + (6 – 1)d 19 = a 1 + 5d a 1 = 19 – 5d 10 = a 1 + (3 – 1)d 10 = a 1 + 2d a 1 = 10 – 2d Example 3

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