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10.2 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Analyze Arithmetic Sequences and Series

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10.2 Warm-Up 1.0, 3, 6, 9, 12, … 2.13, 8, 3, –2, –7, … ANSWER Each is 3 more than the previous term. ANSWER Each is 5 less than the previous term. How is each term in the sequence related to the previous term?

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10.2 Warm-Up 3.3, 6, 9, 12, … 4. –8, –16, –24, –32, … ANSWER a n = 3n; a 5 = 15 ANSWER a n = –8n; a 5 = –40 Write a rule for the n th term of the sequence. Then find a 5.

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10.2 Example 1 Tell whether the sequence is arithmetic. a. –4, 1, 6, 11, 16,... b. 3, 5, 9, 15, 23,... SOLUTION Find the differences of consecutive terms. a 2 – a 1 = 1 – (–4) = 5 a. a 3 – a 2 = 6 – 1 = 5 a 4 – a 3 = 11 – 6 = 5 a 5 – a 4 = 16 – 11 = 5 b. a 2 – a 1 = 5 – 3 = 2 a 3 – a 2 = 9 – 5 = 4 a 4 – a 3 = 15 – 9 = 6 a 5 – a 4 = 23 – 15 = 8

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10.2 Each difference is 5, so the sequence is arithmetic. ANSWER The differences are not constant, so the sequence is not arithmetic. Example 1

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10.2 Guided Practice 1. Tell whether the sequence 17, 14, 11, 8, 5,... is arithmetic. Explain why or why not. ANSWER Arithmetic; There is a common differences of –3.

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10.2 a. 4, 9, 14, 19,... b. 60, 52, 44, 36,... SOLUTION The sequence is arithmetic with first term a 1 = 4 and common difference d = 9 – 4 = 5. So, a rule for the nth term is: a n = a 1 + (n – 1) d = 4 + (n – 1)5 = –1 + 5n Write general rule. Substitute 4 for a 1 and 5 for d. Simplify. The 15th term is a 15 = –1 + 5(15) = 74. Write a rule for the nth term of the sequence. Then find a 15. a. Example 2

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10.2 The sequence is arithmetic with first term a 1 = 60 and common difference d = 52 – 60 = –8. So, a rule for the nth term is: a n = a 1 + (n – 1) d = 60 + (n – 1)(–8) = 68 – 8n Write general rule. Substitute 60 for a 1 and – 8 for d. Simplify. b. The 15th term is a 15 = 68 – 8(15) = –52. Example 2

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10.2 One term of an arithmetic sequence is a 19 = 48. The common difference is d = 3. a n = a 1 + (n – 1)d a 19 = a 1 + (19 – 1)d 48 = a (3) Write general rule. Substitute 19 for n Solve for a 1. So, a rule for the n th term is: a. Write a rule for the nth term. b. Graph the sequence. –6 = a 1 Substitute 48 for a 19 and 3 for d. SOLUTION a. Use the general rule to find the first term. Example 3

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10.2 a n = a 1 + (n – 1)d = –6 + (n – 1)3 = –9 + 3n Write general rule. Substitute –6 for a 1 and 3 for d. Simplify. Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on a line. This is true for any arithmetic sequence. b. Example 3

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10.2 Two terms of an arithmetic sequence are a 8 = 21 and a 27 = 97. Find a rule for the nth term. SOLUTION STEP 1 Write a system of equations using a n = a 1 + (n – 1)d and substituting 27 for n (Equation 1 ) and then 8 for n (Equation 2 ). Example 4

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10.2 STEP 2 Solve the system. 76 = 19d 4 = d 97 = a (4) Subtract. Solve for d. Substitute for d in Equation 1. –7 = a 1 Solve for a 1. STEP 3 Find a rule for a n. a n = a 1 + (n – 1)d Write general rule. = –7 + (n – 1)4 Substitute for a 1 and d. = –11 + 4n Simplify. a 27 = a 1 + (27 – 1)d 97 = a d a 8 = a 1 + (8 – 1)d 21 = a 1 + 7d Equation 1 Equation 2 Example 4

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10.2 Write a rule for the nth term of the arithmetic sequence. Then find a , 14, 11, 8,... ANSWER a n = 20 – 3n; –40 3. a 11 = –57, d = –7 ANSWER a n = 20 – 7n; – a 7 = 26, a 16 = 71 ANSWER a n = –9 + 5n; 91 Guided Practice

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10.2 SOLUTION a 1 = 3 + 5(1) = 8 a 20 = 3 + 5(20) =103 S 20 = 20 ( ) = 1110 Identify first term. Identify last term. Write rule for S 20, substituting 8 for a 1 and 103 for a 20. Simplify. ANSWER The correct answer is C. Example 5

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10.2 You are making a house of cards similar to the one shown. Write a rule for the number of cards in the nth row if the top row is row 1. a. What is the total number of cards if the house of cards has 14 rows? b. House Of Cards Example 6

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10.2 SOLUTION Starting with the top row, the numbers of cards in the rows are 3, 6, 9, 12,.... These numbers form an arithmetic sequence with a first term of 3 and a common difference of 3. So, a rule for the sequence is: a. a n = a 1 + (n – 1) = d = 3 + (n – 1)3 = 3n Write general rule. Substitute 3 for a 1 and 3 for d. Simplify. Example 6

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10.2 SOLUTION Total number of cards = S 14 = 14 ( ) a 1 + a 14 2 = 14 ( ) = 315 Find the sum of an arithmetic series with first term a 1 = 3 and last term a 14 = 3(14) = 42. b. Example 6

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Find the sum of the arithmetic series (2 + 7i). 12 i = 1 ANSWER S 12 = WHAT IF? In Example 6, what is the total number of cards if the house of cards has 8 rows? ANSWER 108 cards Guided Practice

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Is the sequence 2, 103, 204, 305, 406,... arithmetic? Explain your answer. Write a rule for the nth term of the sequence 5, 2, –1, –4,.... Then find a Two terms of an arithmetic sequence are a 5 = 14 and a 30 = 89. Find a rule for the nth term. 3. ANSWERYes; the common difference is 101. ANSWER a n = 8 – 3n ; a 5 = –7 ANSWER a n = –1 + 3n Lesson Quiz

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How many seats are in the theater? A movie theater has 24 seats in the first row and each successive row contains one additional seat. There are 30 rows in all. ANSWER 1155 seats 4. Write a rule for the number of seats in the nth row. ANSWER a n = 23 + n Lesson Quiz

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