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MATHPOWER TM 10, WESTERN EDITION Chapter 2 Number Patterns 2.5 2.5.1

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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 the Common Difference. 2.5.2 Arithmetic Sequences

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-3 + (1)4 -3 + 4 + 4 -3 + (2)4 -3 + 4 + 4 + 4 -3 + (3)4 Continuing with this pattern, the general term is derived as: t n = a + (n - 1) d -3 15 9 a a + da + 2d a + 3d -3 + 4 2.5.3 Arithmetic Sequences

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

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Given the sequence -5, -1, 3, …: a) Find the common difference. d = t 2 - t 1 = (-1) - ( -5) = 4 Note: The common difference may be found by subtracting any two consecutive terms. b) Find t 10. t n = a + (n - 1) d c) Find the general term. t n = a + (n - 1) d d) Which term is equal to 63? t n = a + (n - 1) d 63 = - 5 + 4n - 4 72 = 4n 18 = n t 18 = 63 a = -5 n = 10 d = 4 t n = ? a = -5 n = ? d = 4 a = -5 n = ? d = 4 t n = 63 t 10 = -5 + (10 - 1) 4 = -5 + (9) 4 t 10 = 31 = -5 + (n - 1) 4 = -5 + 4n - 4 t n = 4n - 9 63 = -5 + (n - 1) 4 2.5.5 Finding the Terms of an Arithmetic Sequence

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Find the number of terms in 7, 3, -1, - 5 …, -117. t n = a + (n - 1) d 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? t n = a + (n - 1) d a = 7 n = ? d = -4 t n = - 117 a = 65 n = ? d = - 6 t n = 11 -117 = 7 + (n - 1) (-4) -117 = 7 - 4n + 4 -117 = -4n + 11 -128 = -4n 32 = n 11 = 65 + (n - 1) (-6) -60 = -6n 10 = n a = 65 n = ? d = - 6 t n = 5 5 = 65 + (n - 1) (-6) -66 = -6n n = 11 2.5.6 Finding the Number of Terms of an Arithmetic Sequence The 10th row contains 11 bricks.There are 11 rows in this pile. There are 32 terms in the sequence.

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Arithmetic means are 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 t n = a + (n - 1)d Therefore, the terms are: a = 6 n = 7 d = ? t n = 30 30 = 6 + (7 - 1)d 30 = 6 + 6d 24 = 6d 4 = d 6,, 30 10,14,18,22, 26 2.5.7 Arithmetic Means

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

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