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Day 1: Introduction to Sequences Alg 2X Unit 10 Sequences and Series Example: 1, 3, 5, 7, 9, 11, … A _____________________ is an ordered set of numbers. Each number is called a _________________ of the sequence. What is the 1st term? _______ 3rd term? _________ 6th term? ________ 10th term? __________ The sequence above is called an _______________ sequence because it goes on forever (notice the …). If the sequence ends (ex: 2, 4, 6, 8), then it is called a _______________ sequence. Sometimes it is helpful to list your sequence in a table to help you map the number of the term to the term itself. Term Number Term Value Normally, we will use sequence notation: n and an . n represents _____________________ and an represents _____________________ So for example, a4 7 means “The _________ term is equal to ________” Examples: Use the sequence : a) a2 _____ 2, 6, 10, 14, 18, 22, 26 b) a5 _____ c) a___ 10 d) a___ 14 d) a7 _____ Ok, now that we understand the notation, let’s talk a little more abstractly about sequences. What is the term previous to an ? __________. What is the next term after an ? ___________. What is the next term after an 3 ? ___________. What is the term previous to an 3 ? ___________. RECURSIVE FORMULA vs EXPLICIT FORMULA: ____________ Formula – _____________________________ (MUST identify one term; often includes “an-1”) ____________ Formula – _____________________________ (often includes “n”) EXAMPLE: 3, 6, 9, 12, 15, … RECURSIVE FORMULA: EXPLICIT FORMULA: Recursive Formula Problems: 1. a1 = 5 and an 2(an1 ) 3 Find the first 5 terms in the sequence. 2. Find the first 5 terms in the sequence with a1 = - 2 and an 3(an1 ) 2 3. Find the first 5 terms in the sequence which has a1 7 and an 2(an1 ) 4 4. What is your first thought if I ask you to find the 1000th term of the sequence in #3? Another way to explain EXPLICIT FORMULA: An explicit formula defines the nth term in a sequence as a function of n. Example: Find the first 5 terms of the sequence an 2(n 5) 3 . Find the 1000th term of the same sequence: Explicit Formula Problems: 1. Find the first 5 terms of the sequence an n 2 2n . Then find the 25th term and the 100th term. 2. Find the first 5 terms of the sequence an 3n 5 . Then find a25 and a100. 3. Find the 3rd, 6th, 10th and 17th terms in the sequence an 3n 1 FINDING FORMULAS: Find an explicit formula for each sequence: You may use the table to help. 1. 3, 6, 9, 12, 15, 18, … Term Number Term Value Formula: an = 2. 1 2 3 4 5 , , , , 2 3 4 5 6 Term Number Term Value Formula: 3. an = 1, 4, 9, 16, 25, 36 … Term Number Term Value Formula: 4. an = 4, 9, 16, 25, 36, 49 … Term Number Term Value Formula: an = Quick Wrap-Up: Find a5 for each sequence below: a) a1 3 ; an an1 4 b) an 8n 7