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Precalculus Notes on Introduction to Sequences I. Starting Activity Describe in words the pattern for each list of numbers. 1. 1, 4, 7, 10, … 2. 5, -1, -7, -13, … Add 3 Arithmetic 4. 9/16/13 HW: Pg 662 #1 – 11 odd 3. 1, 3, 9, 27, 81, … 3n1 where n = 1 to start Geometric Subtract 6 Arithmetic 64, -32, 16, -8, 4, … 5. 2 3 4 5 6 , , , , ,… 1 2 3 4 5 6. 61, 52, 63, 94, 46, … n 1 n = 1 to start n Divide by -2 The perfect squares backwards Multiply by -1/2 Geometric 7. 1, 1, 2, 3, 5, 8, 13, … The sum of the previous two terms Recursive 8. 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, … If even divide by 2, If odd multiply by 3 and add 1 II. Introduction to Sequences A. A sequence is a function that computes an ordered list where the domain is the set of natural numbers. B. The elements in the range of a sequence, called the terms of the sequence, are 𝒂𝟏 , 𝒂𝟐 , 𝒂𝟑 , …. C. Instead of using function notation, it is customary to use 𝒂𝒏 where 𝒂𝒏 = 𝒇(𝒏). The letter n is used instead of x as a reminder that n represents a natural number. D. The elements of both the domain and the range of a sequence are ordered. E. A sequence is a finite sequence if the domain is the set {1, 2, 3, 4, …, n}, where n is a natural number. i. Example: the sequence of the days in June: 1, 2, 3, 4, …, 27, 28, 29, 30 ii. Create your own example. F. A sequence is an infinite sequence if the domain is the set of all natural numbers: {1, 2, 3, …, n, …}. i. Example: the sequence of the natural numbers multiplied by 2: 2, 4, 6, 8, … G. To find any term of a sequence, plug the term number in for n and calculate. i. Example: given an 5n 1 , find the first two terms. n 1; an 5(1) 1 4 n 2; an 5(2) 1 9 III. Introduction to Arithmetic Sequences A. A banquet hall uses tables that seat 4, one person on each side. For a large party, the tables are positioned end to end in a long row. Two tables will seat 6, three tables will seat 8, and four tables will seat 10. How many tables should be set end to end to seat 40? Tables People 1 4 2 6 3 8 4 10 … … n 40 The common difference is 2 (this is the same as the slope of a linear function) an 2 n c an 4 2(n 1) 4 2(1) c an 2n 2 c2 40 2n 2 n 19 40 2n 2 n 19