MATH 2B Lecture 17: Sequences
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Math 2b lecture 17 11. 1: sequences. Given some sequences, find the recursive and explicit forms: (cid:1853)(cid:2869),(cid:1853)(cid:2870),(cid:1853)(cid:2871),(cid:1853)(cid:2872), ,(cid:1853),(cid:1853)+(cid:2869), . Sequences are defined explicitly ((cid:1853)=(cid:1866)) or are defined recursively ((cid:1853)=(cid:1858)(cid:4666)(cid:1853) (cid:2869)(cid:4667),(cid:1853)(cid:2869)=#(cid:4667: (cid:883),(cid:884),(cid:885),(cid:886),(cid:887),(cid:888), . Explicit: (cid:1853)=(cid:1866)(cid:2870: (cid:1005),(cid:1008),(cid:1013),(cid:1005)(cid:1010),(cid:1006)(cid:1009),(cid:1007)(cid:1010), , (cid:1005),(cid:1006),(cid:1008),(cid:1012),(cid:1005)(cid:1010),(cid:1007)(cid:1006), , (cid:1007),(cid:1009),(cid:1011),(cid:1013),(cid:1005)(cid:1005),(cid:1005)(cid:1007), . Fear not, there are formulas you can follow depending on the type of sequence we have. Geometric sequences: add the same amount each time, a) and b) are examples, (cid:1853)(cid:2869),(cid:1853)(cid:2869)+(cid:1856),(cid:1853)(cid:2869)+(cid:884)(cid:1856),(cid:1853)(cid:2869)+(cid:885)(cid:1856), . Check a) and b) now that you know the equation: multiply by the same # each time, c) is an example, (cid:1853)(cid:2869),(cid:1853)(cid:2869) (cid:1870),(cid:1853)(cid:2869) (cid:1870)(cid:2870),(cid:1853)(cid:2869) (cid:1870)(cid:2871), . Find the explicit formula for the sequence: 5, -1, -7, -(cid:1005)(cid:1007), (cid:1853)(cid:2869)=(cid:887) (cid:1856)= (cid:888) (cid:1853)=(cid:887)+(cid:4666) (cid:888)(cid:4667)(cid:4666)(cid:1866) (cid:883)(cid:4667)=(cid:887) (cid:888)(cid:1866)+(cid:888)=(cid:2778)(cid:2778) . If a sequence is bounded and monotone, then is must be convergent. If a sequence is monotone, it is always non-decreasing or always non-increasing: a sequence is called bounded if it is bounded both below and above.