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8.1: Sequences Craters of the Moon National Park, Idaho Photo by Vickie Kelly, 2008 Greg Kelly, Hanford High School, Richland, Washington A sequence is a list of numbers written in an explicit order. an a1, a2 , a3, ... , an , ... nth term Any real-valued function with domain a subset of the positive integers is a sequence. If the domain is finite, then the sequence is a finite sequence. In calculus, we will mostly be concerned with infinite sequences. A sequence is defined explicitly if there is a formula that allows you to find individual terms independently. an Example: 1 n n2 1 To find the 100th term, plug 100 in for n: 1 100 a100 1 1002 1 10001 A sequence is defined recursively if there is a formula that relates an to previous terms. Example: b1 4 bn bn1 2 for all n 2 We find each term by looking at the term or terms before it: b1 4 b2 b1 2 6 b3 b2 2 8 b4 b3 2 10 You have to keep going this way until you get the term you need. An arithmetic sequence has a common difference between terms. Example: 5, 2, 1, 4, 7, ... ln 2, ln 6, ln18, ln 54, ... Arithmetic sequences can be defined recursively: or explicitly: d 3 6 d ln 6 ln 2 ln ln 3 2 an an 1 d an a1 d n 1 An geometric sequence has a common ratio between terms. Example: 1, 2, 4, 8, 16, ... 102 , 101 , 1, 10, ... Geometric sequences can be defined recursively: or explicitly: r 2 101 r 2 10 10 an an1 d an a1 d n 1 If the second term of a geometric sequence is 6 and the fifth term is -48, find an explicit rule for the nth term. Example: a1 r 48 a1 r 6 4 r 3 8 r 2 a2 a1 r 21 6 a1 2 3 a1 an 3 2 n 1 You can determine if a sequence converges by finding the limit as n approaches infinity. 2n 1 Does an converge? n 2n 1 lim n n 2n 1 lim n n n 2n 1 lim lim n n n n 20 2 The sequence converges and its limit is 2. Absolute Value Theorem for Sequences If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero. Don’t forget to change back to function mode when you are done plotting sequences. p