1. MATHPOWER TM 12, WESTERN EDITION 6.1 6.1.1 Chapter 6 Sequences and Series

2. 6.1.2 Sequences A sequence is an ordered list of numbers usually separated by commas. A recursive formula relates each term of a sequence to the term before it. Write the first four terms of the sequence defined by t 1 = 5, t n = 3t n - 1 + n, n > 1. t 1 = 5 t 2 = 3t 1 + 2 = 3(5) + 2 = 17 t 3 = 3t 2 + 3 = 3(17) + 3 = 54 t 4 = 3t 3 + 4 = 3(54) + 4 = 166 The first four terms are 5, 17, 54, and 166.

3. 6.1.3 Writing a Recursive Formula Write a recursive formula for each sequence: a) 1, 6, 11, 16, 21,... t 1 = 1,t n = t n - 1 + 5,n > 1 b) 1, 5, 14,... t 1 = 1,t n = t n - 1 + n 2,n > 1 c) 3, 7, 13, 21,... t 1 = 3,t n = t n - 1 + 2n,n > 1 Suggested Questions: Page 290 1-15 odd, 19-27 odd

4. Arithmetic Sequences An arithmetic sequence is a sequence that has a constant common difference, d, between successive terms. For an arithmetic sequence, if the first term t 1, is a and the common difference is d, then t n = a + (n - 1)d. 6.1.4 Example: In an arithmetic sequence, the third term is 11 and the eighth term is 46. Find the first three terms of the sequence. t n = a + (n - 1)d t 3 = 11 11 = a + 2d t 8 = 46 46 = a + 7d a + 7d = 46 a + 2d = 11 5d = 35 d = 7 a = -3 Therefore, the first three terms are -3, 4, and 11.

5. 6.1.5 Arithmetic Series An arithmetic series is the sum of the terms of an arithmetic sequence. For the general arithmetic series, the sum of the first n terms is represented by the following formulas:

6. 6.1.6 Arithmetic Series Find the sum of the sequence 17, 12, 7,..., -38. t n = a + (n - 1)d -38 = 17 + (n - 1)(-5) -38 = 17 - 5n + 5 -60 = -5n 12 = n S 12 = -126 The sum of the sequence is -126.

7. 6.1.7 Summation Notation The Greek letter  (sigma) corresponds to the English letter S and stands for “sum”. Sigma notation is used to abbreviate the writing of a series.  7 k = 3 (5k - 3) The upper limit The lower limit The general term of the sequence This is read as “the summation of 3 to 7 of (5k - 3)”.

8. 6.1.8 Summation Notation Write the series in expanded form and then find the sum: = 6(3) + 3 + 6(4) + 3 + 6(5) + 3 + 6(6) + 3 + 6(7) + 3 = 21 + 27 + 33 + 39 + 45 = 165 Write the series 3 + 6 + 9 + 12 + 15 + 18 in summation notation. There are 6 terms in the series. t n = a + (n - 1)d = 3 + (n - 1)3 = 3n The series written in summation notation is

9. 6.1.9 Suggested Questions: Pages 295 and 296 1, 3, 9, 18, 21, 25, 27, 32, 36