SECTION 12.2 Arithmetic Sequences 873 Creating a Floor Design A ceramic tile floor is designed in the shape of a trapezoid 20 feet wide at the base and 10 feet wide at the top. See Figure 14. The tiles, which measure 12 inches by 12 inches, are to be placed so that each successive row contains one fewer tile than the preceding row. How many tiles will be required? Solution EXAMPLE 8 The bottom row requires 20 tiles and the top row, 10 tiles. Since each successive row requires one fewer tile, the total number of tiles required is = + + + + + S 20 19 18 11 10 This is the sum of an arithmetic sequence; the common difference is −1. The number of terms to be added is = n 11, with the first term = a 20 1 and the last term = a 10. 11 The sum S is ( ) ( ) = + = + = S n a a 2 11 2 20 10 165 1 11 In all, 165 tiles will be required. Figure 14 Concepts and Vocabulary 12.2 Assess Your Understanding 1. In a(n) sequence, the difference between successive terms is a constant. 2. True or False For an arithmetic sequence { } an whose first term is a1 and whose common difference is d, the n th term is determined by the formula = + a a nd. n 1 3. If the 5th term of an arithmetic sequence is 12 and the common difference is 5, then the 6th term of the sequence is . 4. True or False The sum Sn of the first n terms of an arithmetic sequence { } an whose first term is a1 is found using the formula ( ) = + S n a a 2 . n n 1 5. Multiple Choice An arithmetic sequence can always be expressed as a(n) sequence. (a) Fibonacci (b) alternating (c) increasing (d) recursive 6. Multiple Choice If = − + a n2 7 n is the n th term of an arithmetic sequence, the first term is . (a) −2 (b) 0 (c) 5 (d) 7 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Skill Building In Problems 7–16, show that each sequence is arithmetic. Find the common difference, and list the first four terms. 7. { } { } = + s n 4 n 8. { } { } = − s n 5 n 9. { } { } = − a n2 5 n 10. { } { } = + b n3 1 n 11. { } { } = − c n 6 2 n 12. { } { } = − a n 4 2 n 13. { } { } = − t n 1 2 1 3 n 14. { } { } = + t n 2 3 4 n 15. { } { } = s ln3 n n 16. { } { } = s e n n ln In Problems 17–24, find the nth term of the arithmetic sequence { } an whose first term a1 and common difference d are given. What is the 51st term? 17. = = a d 2; 3 1 18. = − = a d 2; 4 1 19. = = − a d 8; 7 1 20. = = − a d 6; 2 1 21. = = a d 0; 1 2 1 22. = = − a d 1; 1 3 1 23. = = a d 2; 2 1 24. π = = a d 0; 1 In Problems 25–30, find the indicated term in each arithmetic sequence. 25. 100th term of 2, 4, 6, . . . 26. 80th term of −1,1,3, . . . 27. 90th term of − − 3, 3, 9, . . . 28. 80th term of − 5, 0, 5, . . . 29. 80th term of 2, 5 2 ,3, 7 2 , . . . 30. 70th term of 2 5, 4 5, 6 5, . . . In Problems 31–38, find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 31. 8th term is 8; 20th term is 44 32. 4th term is 3; 20th term is 35 33. 9th term is −5; 15th term is 31 34. 8th term is 4; 18th term is −96 35. 15th term is 0; 40th term is −50 36. 5th term is −2; 13th term is 30 37. 14th term is −1; 18th term is −9 38. 12th term is 4; 18th term is 28
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