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IB SL/HL 1.1 IB Diploma Mathematics SL/HL Sequences and Series Ms. Rola AbuSager Please refer to the powerpoint presentation on Sequences and Series before reading the lessons. Lesson 1 Concept of sequences and series Form of arithmetic sequences nth term of arithmetic sequences Notes Title: Chapter 2: Sequences and Series See p54 A sequence is an ordered list of numbers. e.g. A: 5, 10, 15, 20, … B: 2, 6, 18, 54, … C: 7, 4, 10, 3, … Some sequences have a pattern; some do not. Sequences can be finite or infinite. Each number in a sequence is call a term. In sequence A, 5 is the first term, 10 is the second term etc. If there is a pattern, we can sometimes write an expression for a general, nth term. The nth term in sequence A is 5n. The letter u is used to represent a term. So, for sequence A, u1 = 5 u2 = 10 un = 5n A series is a sequence with the terms added together. e.g. 5 + 10 + 15 + 20 + …. Many sequences and series have a common structure. We will study two: arithmetic and geometric. ***** Subtitle: Arithmetic Sequences If the same number is added each time to get to the next term in a sequence then the sequence is arithmetic (also called an arithmetic progression). Sequence A above is an arithmetic sequence. Such sequences are defined by the first term, a, and the common difference, d. e.g.1. e.g.2. In A, u1 = 5 and d = 5 In the sequence 4, 7, 10, 13, … u1 = 4 and d = 3 For all arithmetic sequences we have the general formula un = u1 + (n – 1) d See examples 2-5 on p.57-58 Assignment #1: Pg. 58 #1a to 1e all and 2a to 2d Due: Wednesday Sept. 30th, 2009 To: email address: [email protected] or [email protected] Please email me if you have any questions! IB SL/HL 1.1 Lesson 2 Finite sum of an arithmetic series Introductory Exercise Calculate the sum of the integers from 1 to 100. Notes Title: The Finite Sum of an Arithmetic Series, Sn See pg. 66-67 It can be shown that the sum of a finite number of terms of an arithmetic series is equal to half the number of terms × (first term + last term) Algebraically, the sum of the first n terms is Sn = n (u1 + un) 2 or n ( u1+ u1+ (n-1)d) 2 n Sn = (2u1 + (n-1)d) 2 Sn = E.g. An arithmetic sequence has 5th term 7 and common difference 3. Find the sum of the first 12 terms. Method Find u1 n (2 u1 + (n-1)d) 2 Answer: 138 Use Sn = Assignment #2: Pg. 67 Exercise 2E.2 #1a to 1d and 2a to 2c Due: Thursday Oct. 1st. , 2009 To: email address: [email protected] or [email protected] Please email me if you have any questions! IB SL/HL 1.1 Lesson 3 Sigma notation (not limited to arithmetic series or i = 1 to ….) Another way to express a sum is to use sigma notation. (sigma) is the first letter of the Greek word for ‘sum’ 100 E.g.1 1 + 2 + 3 + 4 + …. + 100 can be written as i i 1 Read as, “the sum as i goes from 1 to 100 of i”. E.g.2 6 + 11 + 16 + 21 + ….. + 36 can be written as 7 (1 5n) n 1 The “counting” letter is called the index. i, n and k are commonly used. The index does 6 not have to start at 1. The series in example 2 can also be written as (6 5n) , for n 0 example. Of course, sigma notation can be used to represent other kinds of series, not just arithmetic. Example Write the series below using sigma notation. 2 + 5 + 10 + 17 + …. + 82 9 Answer: (n 2 1) n 1 Assignment #3: Pg. 68 #3.7.11. & 13 Due: Saturday Oct. 3rd. , 2009 To: email address: [email protected] or [email protected] Please email me if you have any questions! Good Luck!