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Chapter 9-1 Introduction to Sequences Obj: To find the nth term of a sequence, and to write rules for sequences. Who uses this? β Sequences are used to model many natural phenomenon, such as changes in rabbit population In 1202, Italian Mathematician Leonardo Fibonacci described how fast rabbits breed under ideal circumstances. Fibonacci noted the number of pairs of pairs of rabbits each month and formed a famous pattern called the Fibonacci sequence. Sequences A sequence is an ordered set of numbers. Each number in the sequence is called a term of the sequence. Infinite Sequence β continues without end, unlimited number of terms (ex. Natural Numbers) Finite Sequence β has a limited number of terms. One way to consider a sequence is with sequential natural numbers as the Domain, and the terms of the sequence as the Range. Term number: Term value: 1 1 2 5 3 9 4 13 5 17 6 7 8β¦ ____ ____ ____ Values in the domain are called Term Numbers, and are represented by n. Instead of functional notation, such as a(n), sequence values are written by using subscripts. The first term is π1 , the second term is π2 , and the nth term is ππ . Because a sequence is a function, each number βnβ has only one term value associated with ππ . Recursive Formula β is a rule in which one or more previous terms are used to generate the next terms. (ex The Fibonacci Sequence is expressed by the rule: π1 = 1, π2 = 1, πππ ππ = ππβ2 + ππβ1 , π€βπππ π β₯ 3. Ex 1. Finding Terms of a Sequence by using a Recursive Formula: Find the first five terms of the sequence with: a) π1 = 5 πππ ππ = 2ππβ1 + 1 πππ π β₯ 2. (Hint: make a table) π 2ππβ1 + 1 ππ 1 Given 5 2 2(5) + 1 3 4 5 b) π1 = β3 πππ ππ = ππβ1 β 8 Ex 2. Finding Terms of a Sequence by Using an Explicit Formula Find the first five terms of the sequence with: π) ππ = 2π β 3 (againβ¦ a table helps) π) ππ = π2 β 2π Writing Rules for sequences TIP: Linear patterns β have constant first differences Quadratic patterns β have constant second differences Exponential patterns β have constant ratios Write a possible explicit rule for the nth term of each sequence π) 7, 5, 3, 1, β1, β¦ Terms: 7 (Write the terms and try to find a common difference) 5 3 1 -1 The first differences are constant so the sequence is linear. You tryβ¦ π) 25, 15, 5, β5, 15, β¦ Terms: 25 15 Common difference? 5 -5 -15 12 24 48 π) 3, 6, 12, 24, 48, β¦ Terms: 3 6 Homework p.629/ 16-24 all find common differences and ratios