NAME 10-2 DATE PERIOD Study Guide and Intervention Arithmetic Sequences and Series Arithmetic Sequences Definition d = an + 1 - an The common difference in an arithmetic sequence with consecutive terms … 5, 7, … is 7 - 5 = 2. an = a1 + (n - 1) d where a1 is the common difference and n is any positive integer. The fifth term of the arithmetic sequence with first term 3 and common difference 2 is 3 + (4 × 2) = 11. Common Difference nth Term of an Arithmetic Sequence Example Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Find the thirteenth term of the arithmetic sequence with a1 = 21 Example 1 and d = - 6. Use the formula for the nth term of an arithmetic sequence with a1 = 21, n = 13, and d = – 6. an = a1 + (n - 1) d Formula for the nth term a13 = 21 + (13 – 1) (-6) n = 13, a1 = 21, d = - 6 a13 = -51 Example 2 Write an equation for the nth term of the arithmetic sequence -14, -5, 4, 13, … . In this sequence, a1 = -14 and d = 9. Use the formula for an to write an equation. an = a1 + (n - 1) d Formula for the nth term an = –14 + (n - 1)(9) a1 = -14, d = 9 an = –14 + 9n - 9 Distributive Property an = 9n - 23 Simplify. Exercises Find the indicated term of each arithmetic sequence. 1. Find the twentieth term of the arithmetic sequence with a1 = 15 and d = 4. 2. Find the seventh term of the arithmetic sequence with a1 = -81 and d = 12. 3. Find the eleventh term of the arithmetic sequence with a1 = 42 and d = – 5. 4. Find a31 of the arithmetic sequence 18, 15, 12, 9, …. 5. Find a100 of the arithmetic sequence -63, -58, -53, -48, .... Write an equation for the nth term of each arithmetic sequence. 6. a1 = 15 and d = 38 7. a1 = 72 and d = -13 8. -56, -39, -22, -5, … 9. -94, -52, -10, 32, … 10. 63, 70, 77, 84, … Chapter 10 11 Glencoe Algebra 2 Lesson 10-2 Term NAME 10-2 DATE PERIOD Study Guide and Intervention (continued) Arithmetic Sequences and Series Arithmetic Series A shorthand notation for representing a series makes use of the 5 Greek letter Σ. The sigma notation for the series 6 + 12 + 18 + 24 + 30 is ∑ 6n. n=1 Partial Sum of an Arithmetic Series The sum Sn of the first n terms of an arithmetic series is given by the formula n n [2a + (n - 1)d ] or S = − Sn = − (a1 + an). 1 n 2 2 Example 2 Example 1 Find Sn for the arithmetic series with a1 = 14, an = 101, and n = 30. k=1 Use the sum formula for an arithmetic series. n (a1 + an) Sum formula Sn = − S30 18 Evaluate ∑ (3k + 4). 2 30 = − (14 + 101) n = 30, a1 = 14, an = 101 2 = 15(115) Simplify. = 1725 Multiply. The sum of the series is 1725. The sum is an arithmetic series with common difference 3. Substituting k = 1 and k = 18 into the expression 3k + 4 gives a1 = 3(1) + 4 = 7 and a18 = 3(18) + 4 = 58. There are 18 terms in the series, so n = 18. Use the formula for the sum of an arithmetic series. n Sn = − (a1 + an) Sum formula S18 2 18 = − (7 + 58) 2 = 9(65) = 585 n = 18, a1 = 7, an = 58 Simplify. Multiply. 18 So ∑ (3k + 4) = 585. k=1 Find the sum of each arithmetic series. 1. a1 = 12, an = 100, n = 12 2. a1 = 50, an = -50, n = 15 3. a1 = 60, an = -136, n = 50 4. a1 = 20, d = 4, an = 112 5. a1 = 180, d = -8, an = 68 6. a1 = -8, d = -7, an = -71 7. a1 = 42, n = 8, d = 6 1 8. a1 = 4, n = 20, d = 2 − 2 10. 8 + 6 + 4 + … + -10 42 12. ∑ (4n - 9) n = 18 Chapter 10 9. a1 = 32, n = 27, d = 3 11. 16 + 22 + 28 + … + 112 50 44 13. ∑ (3n + 4) 14. ∑ (7j - 3) j=5 n = 20 12 Glencoe Algebra 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Exercises

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