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Arithmetic Sequences

Chapter 2 Number Patterns. 2.5. Arithmetic Sequences. 2.5. 1. MATHPOWER TM 10, WESTERN EDITION. Arithmetic Sequences. An Arithmetic Sequence is a sequence where each term is formed from the preceding term by adding a constant to the preceding term.

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Arithmetic Sequences

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  1. Chapter 2 Number Patterns 2.5 Arithmetic Sequences 2.5.1 MATHPOWERTM 10, WESTERN EDITION

  2. Arithmetic Sequences An Arithmetic Sequence is a sequence where each term is formed from the preceding term by adding a constant to the preceding term. Consider the sequence -3, 1, 5, 9. This sequence is found by adding 4 to the previous term. The constant term which is added to each term to produce the sequence is called theCommon Difference. 2.5.2

  3. Arithmetic Sequences 9 1 5 -3 -3 -3 + 4 -3 + 4 + 4 -3 + (2)4 -3 + 4 + 4 + 4 -3 + (3)4 -3 + (1)4 a + d a + 2d a a + 3d Continuing with this pattern, the general term is derived as: tn = a + (n - 1) d 2.5.3

  4. The General Arithmetic Sequence General Term Number or Position of the Term tn = a + (n - 1) d First Term Common Difference 2.5.4

  5. Finding the Terms of an Arithmetic Sequence Given the sequence -5, -1, 3, …: a) Find the common difference. d = t2 - t1 = (-1) - ( -5) = 4 Note: The common difference may be found by subtracting any two consecutive terms. c) Find the general term . b) Find t10 . tn = a + (n - 1) d a = -5 n = ? d = 4 tn = a + (n - 1) d a = -5 n = 10 d = 4 tn = ? = -5 + (n - 1) 4 = -5 + 4n - 4 tn = 4n - 9 t10 = -5 + (10 - 1) 4 = -5 + (9) 4 t10 = 31 a = -5 n = ? d = 4 tn =63 d) Which term is equal to 63? 63 = - 5 + 4n - 4 72 = 4n 18 = n tn = a + (n - 1) d 63 = -5 + (n - 1) 4 t18 = 63 2.5.5

  6. Finding the Number of Terms of an Arithmetic Sequence Find the number of terms in 7, 3, -1, - 5 …, -117 . tn = a + (n - 1) d a = 7 n = ? d = -4 tn =- 117 -117 = 7 + (n - 1) (-4) -117 = 7 - 4n + 4 -117 = -4n + 11 -128 = -4n 32 = n There are 32 terms in the sequence. A pile of bricks is arranged in rows. The number of bricks in each row forms a sequence 65, 59, 53, …, 5. Which row contains 11 bricks? How many rows are there? tn = a + (n - 1) d tn = a + (n - 1) d a = 65 n = ? d = - 6 tn =5 a = 65 n = ? d = - 6 tn =11 5 = 65 + (n - 1) (-6) -66 = -6n n = 11 11 = 65 + (n - 1) (-6) -60 = -6n 10 = n The 10th row contains 11 bricks. There are 11 rows in this pile. 2.5.6

  7. Arithmetic Means Arithmetic meansare the terms that are between two given terms of an arithmetic sequence. Insert five arithmetic means between 6 and 30. 6 _ _ _ _ _ 30 7 terms altogether tn = a + (n - 1)d a = 6 n = 7 d = ? tn =30 30 = 6 + (7 - 1)d 30 = 6 + 6d 24 = 6d 4 = d Therefore, the terms are: 26 6, , 30 10, 14, 18, 22, 2.5.7

  8. Assignment Suggested Questions: Pages 74 - 76 1 - 43 odd 46, 47, 49, 50 52, 53, 56, 57 2.5.8

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