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Warm-up Describe the pattern and find the next three terms. 1. 2, 3, 5, 8, 12, .... 2. 1/2, 1/4, 1/8, 1/16, 1/32... 3. 5, 2, 6, 3, 9, 6, 14, .... 1 [9.1] SEQUENCES AND SERIES Sequences def: a collection of terms ordered so that it has a first member, second member, etc... a1, a2, a3, a4, . . . , an a1 is the first term! & an is the nth term! Ex.) 2, 3, 5, 8, 12, .... is a sequence. a1 = a3 = 2 Two Types of Sequences a) finite sequence 3, 6, 9, 12, 15 # of terms? b) infinite sequence 3, 6, 9, 12, 15, .... # of terms? 3 Explicit Form - a formula for a sequence where each term is found by plugging in the term number. To find the first term, simply plug in 1 for n because a1 is the first term and the value of n is 1. Ex 1: Write the first five terms of 4 Recursive Sequences: The first term(s) will be given and all other terms will be defined using previous terms. In a recursive sequence, the next term is found by using the term before it. Example: Find the first five terms. 1. a1 = 3, ak + 1 = ak - 4 2. a1 = 15, ak + 1 = 2ak + 10 5 *You Try 1. a1 = -2, ak + 1 = ak Find a3 + 2 2. a1 = 11, ak + 1 = 3ak + 2 Find a4. 6 The most famous recursive sequence is the Fibonacci Sequence: Definition: a0 = 1, a1 = 1, ak = ak-2 + ak-1 , where k≥2 Terms: a0 = 1 a1 = 1 a2 = a2-2 + a2-1= a0 + a1 = 1 + 1 = 2 a3 = a3-2 + a3-1= a1 + a2 = 1 + 2 = 3 a4 = a4-2 + a4-1= a2 + a3 = 2 + 3 = 5 Find a5 and a6. 7 *You Try: Determine if the sequence is explicit form or recursive. Then, write the first 5 terms of the sequence. 1. a1 = 4, ak +1 = ak + 5 2. an = 3(-5)n-1 3. a1 = -4, ak +1 = 3(ak + 2) 8 Series def: the sum of the terms of a sequence a) finite b) infinite 3 + 6 + 9 + 12 + 15 3 + 6 + 9 + 12 + 15 + .... 9 Sigma (or summation) Notation = 10 Ex : Find the sum of the series. a) b) 11 Factorials: 0! = 1 1! = 1 2! = 2 1 = 2 3! = 3 2 1 = 6 7! = 7 6 5 4 3 2 1 = 5040 12 Evaluating factorials: 1. 8! 2! 6! 2. 42! 52! 45! 50! 3. (n + 2)! n! 13 [9.2] Arithmetic Sequences EQ: How do we write a rule for an arithmetic sequence? I. Definition of Arithmetic Sequence: a sequence where the difference between terms is constant *In other words, an arithmetic sequence is a sequence where you add the same number to each term to get the next term. Ex 1: -3, 1, 5, 9, 13, ... What is the difference between the numbers? common difference is d = Ex 2: 2, -5, -12, -19, ... 14 *You try: Determine if the sequence is arithmetic. And, if it is, find the common difference, d. 1.) -10, -6, -2, 0, 2, 6, 10 2.) 5, 11, 17, 23, 29 15 Formula for an Arithmetic Sequence an = dn + c Where d is the common difference and c = a1 - d Notice that arithmetic sequences are linear! 16 Ex 3: In an arithmetic sequence the common difference is 5 and a1 = -3. Write the formula for the sequence and find a15. Ex 4: Write a formula for the arithmetic sequence where a5 = 25 and a12 = 39 17 *You Try: 1.) In an arithmetic sequence the common difference is -2 and a1 = 7. Write the formula for the sequence and find a20. 2.) Write a formula for the arithmetic sequence where a12 = -50 and a20 = -18. 18 EQ: How do you find the sum of an arithmetic series? The Sum of a Finite ARITHMETIC Series The sum of the first n terms of an arithmetic series is: Which of these is arithmetic? How do you know? 27 Σ k=1 27 2k - 3 Σ (2k - 3)2 k=1 19 Ex.) Find the sum of the first 25 terms of the arithmetic series (this is also called the 25th partial sum of the series): 20 + 18 + 16 + 14 + ... 1. Find an using an = dn + c 2. Then, plug into *You try: Find the 30th partial sum of the following series: 11 + 16 + 21 + 26 + ... 20 What if the series is written in sigma notation? Ex: Find the sum of the following series. *You try: What about something like these? 7 Σ k=3 7 -2k + 1 Σ -2k2 + 1 k=3 21 Homework: Pg. 625 #s 1,4,9,26,30,69-84 mult. of 3 Pg. 635 #s 1-8,35,40,44,60,61,68,71 22