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Splash Screen. Find the next four terms of the sequence 2, 5, 8, 11, … . A. 13, 15, 17, 19 B. 14, 17, 20, 23 C. 15, 19, 23, 27 D. 15, 20, 26, 33. 5–Minute Check 1. Find the next four terms of the sequence 3, −9, 27, −81, … . A. 162, –324, 648, –1296 B. 243, –729, 2187, –6561

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Splash Screen

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  1. Splash Screen

  2. Find the next four terms of thesequence 2, 5, 8, 11, … . A.13, 15, 17, 19 B.14, 17, 20, 23 C.15, 19, 23, 27 D.15, 20, 26, 33 5–Minute Check 1

  3. Find the next four terms of the sequence 3, −9, 27, −81, … . A.162, –324, 648, –1296 B.243, –729, 2187, –6561 C.243, 729, 2187, 6561 D.324, –1296, 5184, –20,736 5–Minute Check 2

  4. Find the third term of the sequence a1 = 10, an = –2an–1 + 4. A.a3 = –68 B.a3 = –16 C.a3 = 36 D.a3 = 76 5–Minute Check 3

  5. Find the sixth term of the sequence an = . A.a6 = B.a6 = C.a6 = D.a6 =3 5–Minute Check 4

  6. Which of the following sequences is divergent? A.3, 11, 15, 17, … B.9, 7, 5, 3, … C.20, −5, 1.25, –0.3125, … D.32, 16, 8, 4, … 5–Minute Check 6

  7. You found terms of sequences. • Find nth terms of arithmetic sequences. • Find sums of n terms of arithmetic series (Partial Sums). Then/Now

  8. arithmetic sequence • common difference • arithmetic series Vocabulary

  9. Arithmetic Sequences Determine the common difference and the next four terms of the arithmetic sequence –53, –36, –19, … . First, find the common difference. a2a1 = –36 – (–53) or 17 Find the difference between pairs of consecutive terms to determine the common difference. a3a2 = –19 – (–36) or 17 Example 1

  10. Arithmetic Sequences The common difference is 17. Add 17 to the third term to find the fourth term, and so on. a4 = –19 + 17 = –2 a5 = –2 + 17 = 15 a6 = 15 + 17 = 32 a7 = 32 + 17 = 49 The next four terms are –2, 15, 32, and 49. Answer:17; –2, 15, 32, 49 Example 1

  11. Determine the common difference and find the next four terms of the arithmetic sequence 74, 68, 62, 56, … . A. –6; 50, 44, 38, 32 B. –5; 51, 46, 41, 36 C. –4; 52, 48, 44, 40 D. 6; 62, 68, 74, 80 Example 1

  12. Key Concept 2

  13. Explicit and Recursive Formulas Find both an explicit formula and a recursive formula for the nth term of the arithmetic sequence 14, 3, –8, … . First, find the common difference. a2a1 = 3  14 or 11 Find the difference between pairs of consecutive terms to determine the common difference. a3a2 = 8  3 or 11 Example 2

  14. Explicit and Recursive Formulas For an explicit formula, first find c by subtracting the common difference from the first term: c= 14 – (-11) = 25 Next, plug d and c into the formula: d = -11; c = 25  an = -11n + 25 Now you can find any term in the sequence! Example 2

  15. Explicit and Recursive Formulas For a recursive formula, state the first term a1 and then indicate that the next term is the sum of the previous term an  1 and d. a1 = 14 an = an 1 + (11) or an = an 1 – 11 Answer:an= –11n + 25; a1 = 14,an = an – 1 – 11 Example 2

  16. Find both an explicit formula and a recursive formula for the nth term of the arithmetic sequence 15, 33, 51, … . A. an = 15n; a1 = 15, an = an– 1 + 18 B. an = 15n; a1 = 18, an = an– 1 + 15 C. an = 18n – 3; a1 = 15, an = an– 1 + 18 D. an = 18n; a1 = 15, an = an– 1 + 18 Example 2

  17. nth Terms A. Find the 41stterm of the arithmetic sequence 11, 4, –3, –10, …. Example 3

  18. nth Terms B. Find the first term of the arithmetic sequence for which a44 = 229 and d = 8. For this type of problem, use the following formula: Here, is the term given and is the term you are looking for:    Answer:–115 Example 3

  19. Find the 25th term of the arithmetic sequence 49, 40, 31, …. A. –176 B. –167 C. –158 D. 265 Example 3

  20. Find the third term of the arithmetic sequence for which a35= -226 and d = -5. A. a3 = -56 B. a3 = -61 C. a3 = -66 D. a3 = -71 Example 4

  21. Key Concept 6

  22. Now use the formula to find the sum of the series. Sums of Arithmetic Series A. Find the sum of –3 + 2 + 7 + 12 + … + 157. In this sequence, a1 = –3, an = 157, and d = 2 – (–3) or 5. Use the nth term formula to find the number of terms in the sequence n. an = dn + c nth term of an arithmetic sequence 157 = 5n – 8an = 157, d = 5, c= -8 33 = n Simplify. Example 6

  23. Sum of an arithmetic series formula. n = 33, a1 = 3, an = 157 Sums of Arithmetic Series = 2541 Simplify. Answer:2541 Example 6

  24. Sums of Arithmetic Series B. Find the 17th partial sum of the arithmetic series 53 + 31 + 9 + … . In this sequence, a1 = 53 and d = 31 – 53 or –22. Find the 17th term, then use the sum of an arithmetic series formula to solve: a17 = -299 Answer:–2091 Example 6

  25. C. = (10 + 6) + (11 + 6) + ··· + (32 + 6) Sums of Arithmetic Series = 16 + 17 + ··· + 38 The first term of this series is 16, and the last term is 38. The number of terms is equal to the upper bound minus the lower bound plus one, which is 32 – 10 + 1, or 23. Therefore, a1 = 16, an = 38, and n = 23. Use the formula to find the sum of the series. Example 6

  26. Sum of an arithmetic series formula. n = 23, a1 = 16, an = 38 Sums of Arithmetic Series = 621 Simplify. Answer:621 Example 6

  27. Find . A. 2072 B. 2146 C. 4144 D. 4292 Example 6

  28. End of the Lesson

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