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PRECALCULUS CHAPTER NINE Sequences Vocabulary & Notation: Sequence: an ordered progression of numbers. Term: each number in a sequence is a term First term is a1 Second term is a2 nth term is an The subscripts only denote the position of the term in the sequence. Types of Sequences: Arithmetic: a sequence in which there is a common difference between every pair of successive terms. Example: 5,8,11,14 Geometric: a sequence in which there is a common ratio (or quotient) between every pair 1 1 1 1 of successive terms. Example: , , , ,... 2 4 8 16 1 1 1 1 Infinite: there is an infinite number of terms in the sequence. Example: , , , ,... 2 4 8 16 Finite: there is a finite number of terms in the sequence. Example: 5,8,11,14 Sequences are infinite unless otherwise specified. Explicitly Defined Sequence: A formula is given for any term in the sequence. Example: ak = 2k -5 . find the first 5 terms and the 20th term in the sequence. Recursively Defined Sequence: The first term is given and a rule to obtain each succeeding term from the one preceding it. Example: b1 = 8 and bn = bn-1 - 2 for all n >1 . Find the next 4 terms in the sequence. General formulas for finding terms in a sequence: Arithmetic: an = a1 + (n -1)d ( ) Geometric: an = a1 r n-1 To use these: 1) Determine if the sequence is arithmetic or geometric 2) Find the common difference or common ratio. 3) Substitute into the general formula (simplify the arithmetic formula) Example 1: Find the 20th term of the sequence 55, 49, 43…. Then write a recursive and explicit rule for the sequence. Example 2: Find the 8th term of the sequence explicit rule for the sequence. 1 4 16 , , ,... Then write a recursive and 5 5 5