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Holt Algebra Arithmetic Sequences and Series 12-3 Arithmetic Sequences and Series Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

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Holt Algebra Arithmetic Sequences and Series Warm Up Find the 5th term of each sequence. 1. a n = n a n = 4 – n 3. a n = 3n + 4 Write a possible explicit rule for the nth term of each sequence. 4. 4, 5, 6, 7, 8,…5. –3, –1, 1, 3, 5, … –1 19 a n = n + 3a n = 2n – 5

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Holt Algebra Arithmetic Sequences and Series Find the indicated terms of an arithmetic sequence. Find the sums of arithmetic series. Objectives

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Holt Algebra Arithmetic Sequences and Series arithmetic sequence arithmetic series Vocabulary

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Holt Algebra Arithmetic Sequences and Series The cost of mailing a letter in 2005 gives the sequence 0.37, 0.60, 0.83, 1.06, …. This sequence is called an arithmetic sequence because its successive terms differ by the same number d (d ≠ 0), called the common difference. For the mail costs, d is 0.23, as shown.

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Holt Algebra Arithmetic Sequences and Series Recall that linear functions have a constant first difference. Notice also that when you graph the ordered pairs (n, a n ) of an arithmetic sequence, the points lie on a straight line. Thus, you can think of an arithmetic sequence as a linear function with sequential natural numbers as the domain.

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Holt Algebra Arithmetic Sequences and Series Example 1A: Identifying Arithmetic Sequences Determine whether the sequence could be arithmetic. If so, find the common first difference and the next term. –10, –4, 2, 8, 14, … The sequence could be arithmetic with a common difference of 6. The next term is = 20. –10, –4, 2, 8, 14 Differences

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Holt Algebra Arithmetic Sequences and Series Example 1B: Identifying Arithmetic Sequences Determine whether the sequence could be arithmetic. If so, find the common first difference and the next term. –2, –5, –11, –20, –32, … –2, –5, –11, –20, –32 The sequence is not arithmetic because the first differences are not common. Differences –3 –6 –9 –12

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 1a Determine whether the sequence could be arithmetic. If so, find the common difference and the next term. 1.9, 1.2, 0.5, –0.2, –0.9,... The sequence could be arithmetic with a common difference of – , 1.2, 0.5, –0.2, –0.9 –0.7 Differences The next term would be –0.9 – 0.7 = –1.6.

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Holt Algebra Arithmetic Sequences and Series The sequence is not arithmetic because the first differences are not common. Check It Out! Example 1b Determine whether the sequence could be arithmetic. If so, find the common difference and the next term. Differences

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Holt Algebra Arithmetic Sequences and Series Each term in an arithmetic sequence is the sum of the previous term and the common difference. This gives the recursive rule a n = a n – 1 + d. You also can develop an explicit rule for an arithmetic sequence.

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Holt Algebra Arithmetic Sequences and Series Notice the pattern in the table. Each term is the sum of the first term and a multiple of the common difference. This pattern can be generalized into a rule for all arithmetic sequences.

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Holt Algebra Arithmetic Sequences and Series

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Holt Algebra Arithmetic Sequences and Series Example 2: Finding the nth Term Given an Arithmetic Sequence Find the 12th term of the arithmetic sequence 20, 14, 8, 2, 4,.... Step 1 Find the common difference: d = 14 – 20 = –6.

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Holt Algebra Arithmetic Sequences and Series Example 2 Continued Step 2 Evaluate by using the formula. a n = a 1 + (n – 1)d a 12 = 20 + (12 – 1)(–6) = –46 General rule. Substitute 20 for a 1, 12 for n, and –6 for d. The 12th term is –46. Check Continue the sequence.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 2a Find the 11th term of the arithmetic sequence. –3, –5, –7, –9, … Step 1 Find the common difference: d = –5 – (–3)= –2. Step 2 Evaluate by using the formula. a n = a 1 + (n – 1)d General rule. a 11 = –3 + (11 – 1)(–2) Substitute –3 for a 1, 11 for n, and –2 for d. = –23 The 11th term is –23.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 2a Continued n anan –11–13–15–17–19–21–23–9–7–5 –3 Check Continue the sequence.

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Holt Algebra Arithmetic Sequences and Series Find the 11th term of the arithmetic sequence. Check It Out! Example 2b 9.2, 9.15, 9.1, 9.05, … Step 1 Find the common difference: d = 9.15 – 9.2 = –0.05. Step 2 Evaluate by using the formula. a n = a 1 + (n – 1)d General rule. a 11 = (11 – 1)(–0.05) Substitute 9.2 for a 1, 11 for n, and –0.05 for d. = 8.7 The 11th term is 8.7.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 2b Continued Check Continue the sequence. n anan

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Holt Algebra Arithmetic Sequences and Series Example 3: Finding Missing Terms Find the missing terms in the arithmetic sequence 17,,,, –7. Step 1 Find the common difference. a n = a 1 + (n – 1)d –7 = 17 + (5 – 1)(d) –6 = d General rule. Substitute –7 for a n, 17 for a 1, and 5 for n. Solve for d.

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Holt Algebra Arithmetic Sequences and Series Example 3 Continued Step 2 Find the missing terms using d= –6 and a 1 = 17. a 2 = 17 + (2 – 1)(–6) = 11 a 3 = 17 +(3 – 1)(–6) = 5 = –1 a 4 = 17 + (4 – 1)(–6) The missing terms are 11, 5, and –1.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 3 Find the missing terms in the arithmetic sequence 2,,,, 0. a n = a 1 + (n – 1)d 0 = 2 + (5 – 1)d –2 = 4d Step 1 Find the common difference. General rule. Substitute 0 for a n, 2 for a 1, and 5 for n. Solve for d.

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Holt Algebra Arithmetic Sequences and Series = 1 Check It Out! Example 3 Continued The missing terms are Step 2 Find the missing terms using d= and a 1 = 2.

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Holt Algebra Arithmetic Sequences and Series Because arithmetic sequences have a common difference, you can use any two terms to find the difference.

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Holt Algebra Arithmetic Sequences and Series Find the 5th term of the arithmetic sequence with a 8 = 85 and a 14 = 157. Example 4: Finding the nth Term Given Two Terms Step 1 Find the common difference. a n = a 1 + (n – 1)d a 14 = a 8 + (14 – 8)d a 14 = a 8 + 6d 157 = d 72 = 6d 12 = d Let a n = a 14 and a 1 = a 8. Replace 1 with 8. Simplify. Substitute 157 for a 14 and 85 for a 8.

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Holt Algebra Arithmetic Sequences and Series Example 4 Continued Step 2 Find a 1. a n = a 1 + (n – 1)d 85 = a 1 + (8 - 1)(12) 85 = a = a 1 General rule Substitute 85 for a 8, 8 for n, and 12 for d. Simplify.

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Holt Algebra Arithmetic Sequences and Series Example 4 Continued Step 3 Write a rule for the sequence, and evaluate to find a 5. a n = a 1 + (n – 1)d a n = 1 + (n – 1)(12) a 5 = 1 + (5 – 1)(12) = 49 The 5th term is 49. General rule. Substitute 1 for a 1 and 12 for d. Evaluate for n = 5.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 4a Find the 11th term of the arithmetic sequence. a 2 = –133 and a 3 = –121 a n = a 1 + (n – 1)d a 3 = a 2 + (3 – 2)d –121 = –133 + d d = 12 Step 1 Find the common difference. Let a n = a 3 and a 1 = a 2. Replace 1 with 2. Simplify. Substitute –121 for a 3 and –133 for a 2. a 3 = a 2 + d

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 4a Continued a n = a 1 + (n – 1)d –133 = a 1 + (2 – 1)(12) –133 = a –145 = a 1 Step 2 Find a 1. General rule Substitute –133 for a n, 2 for n, and 12 for d. Simplify.

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Holt Algebra Arithmetic Sequences and Series a n = a 1 + (n – 1)d a 11 = –145 + (n – 1)(12) = –25 The 11th term is –25. Step 3 Write a rule for the sequence, and evaluate to find a 11. General rule. Substitute –145 for a 1 and 12 for d. Evaluate for n = 11. Check It Out! Example 4a Continued a 11 = –145 + (11 – 1)(12)

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 4b Find the 11th term of each arithmetic sequence. a 3 = 20.5 and a 8 = 13 a n = a 1 + (n – 1)d a 8 = a 3 + (8 – 3)d Step 1 Find the common difference. a 8 = a 3 + 5d 13 = d –7.5 = 5d –1.5 = d General rule Let a n = a 8 and a 1 = a 3. Replace 1 with 3. Simplify. Substitute 13 for a 8 and 20.5 for a 3. Simplify.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 4b Continued a n = a 1 + (n – 1)d 20.5 = a 1 + (3 – 1)(–1.5) 20.5 = a 1 – = a 1 Step 2 Find a 1. General rule Substitute 20.5 for a n, 3 for n, and –1.5 for d. Simplify.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 4b Continued a n = a 1 + (n – 1)d a 11 = (n – 1)(–1.5) Step 3 Write a rule for the sequence, and evaluate to find a 11. a 11 = 8.5 The 11th term is 8.5. General rule Substitute 23.5 for a 1 and –1.5 for d. Evaluate for n = 11. a 11 = (11 – 1)(–1.5)

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Holt Algebra Arithmetic Sequences and Series In Lesson 12-2 you wrote and evaluated series. An arithmetic series is the indicated sum of the terms of an arithmetic sequence. You can derive a general formula for the sum of an arithmetic series by writing the series in forward and reverse order and adding the results.

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Holt Algebra Arithmetic Sequences and Series

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Holt Algebra Arithmetic Sequences and Series

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Holt Algebra Arithmetic Sequences and Series Find the indicated sum for the arithmetic series. Example 5A: Finding the Sum of an Arithmetic Series S 18 for (–9) + (–20) +... Find the common difference. d = 2 – 13 = –11 Find the 18th term. a 18 = 13 + (18 – 1)(–11) = –174

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Holt Algebra Arithmetic Sequences and Series Example 5A Continued Check Use a graphing calculator. = 18(-80.5) = –1449 Substitute. Sum formula

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Holt Algebra Arithmetic Sequences and Series These sums are actually partial sums. You cannot find the complete sum of an infinite arithmetic series because the term values increase or decrease indefinitely. Remember!

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Holt Algebra Arithmetic Sequences and Series Example 5B: Finding the Sum of an Arithmetic Series Find 1st and 15th terms. a 1 = 5 + 2(1) = 7 a 15 = 5 + 2(15) = 35 Find S 15. = 15(21) = 315 Find the indicated sum for the arithmetic series.

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Holt Algebra Arithmetic Sequences and Series Example 5B Continued Check Use a graphing calculator.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 5a Find the indicated sum for the arithmetic series. S 16 for (–3)+ … d = 7 – 12 = –5 a 16 = 12 + (16 – 1)(–5) = –63 Find the common difference. Find the 16th term.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 5a Continued = 16(–25.5) = –408 Substitute. Sum formula. Simplify. Find S 16.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 5b Find the indicated sum for the arithmetic series. a 1 = 50 – 20(1) = 30 a 15 = 50 – 20(15) = –250 Find 1st and 15th terms.

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Holt Algebra Arithmetic Sequences and Series = 15(–110) = –1650 Check It Out! Example 5b Continued Find S 15. Substitute. Sum formula. Simplify.

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Holt Algebra Arithmetic Sequences and Series The center section of a concert hall has 15 seats in the first row and 2 additional seats in each subsequent row. Example 6A: Theater Application How many seats are in the 20th row? Write a general rule using a 1 = 15 and d = 2. a n = a 1 + (n – 1)d a 20 = 15 + (20 – 1)(2) = = 53 Explicit rule for nth term Substitute. Simplify. There are 53 seats in the 20th row.

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Holt Algebra Arithmetic Sequences and Series Example 6B: Theater Application How many seats in total are in the first 20 rows? Find S 20 using the formula for finding the sum of the first n terms. There are 680 seats in rows 1 through 20. Formula for first n terms Substitute. Simplify.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 6a What if...? The number of seats in the first row of a theater has 14 seats. Suppose that each row after the first had 2 additional seats. How many seats would be in the 14th row? Write a general rule using a 1 = 14 and d = 2. a n = a 1 + (n – 1)d a 14 = 11 + (14 – 1)(2) = = 37 Explicit rule for nth term Substitute. Simplify. There are 37 seats in the 14th row.

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Holt Algebra Arithmetic Sequences and Series Check It Out! Example 6b How many seats in total are in the first 14 rows? Find S 14 using the formula for finding the sum of the first n terms. There are 336 total seats in rows 1 through 14. Formula for first n terms Substitute. Simplify.

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Holt Algebra Arithmetic Sequences and Series Lesson Quiz: Part I 1. Determine whether the sequence could be arithmetic. If so, find the first difference and the next term. –1, –4, –7, –10, –13, … yes; –3,–16 2. Find the 10th term of the arithmetic sequence –2, –5, –8, –11, –14, … –29 3. Find the missing terms in the arithmetic sequence 15,,,, , 16, Find the 6th term of the arithmetic sequence with a 9 = 64 and a 12 =

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Holt Algebra Arithmetic Sequences and Series Lesson Quiz: Part II 5. Find the indicated sum for – The side section of an auditorium has 12 seats in the first row and 3 additional seats in each subsequent row. How many seats are in the 10th row? How many seats in total are in the first 10 rows? 39; 255

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