280 CHAPTER 5 Number Theory and the Real Number System Exercises Warm Up Exercises In Exercises 1– 6, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. A list of numbers that are related to each other by a rule is called a(n) ________. Sequence 2. The numbers that form a sequence are called its ________. Terms 3. A sequence in which each term differs from the preceding term by a constant amount is called a(n) ________ sequence. Arithmetic 4. The amount by which each pair of successive terms differ in an arithmetic sequence is called the common ________. Difference 5. A sequence in which the ratio of any term to the term that directly precedes it is a constant is called a(n) ________ sequence. Geometric 6. The constant determined by dividing any term in a geometric sequence by the term that directly precedes it is called the common ________. Ratio Practice the Skills In Exercises 7–12, write the first five terms of the arithmetic sequence with the first term, a ,1 and common difference, d. 7. a d 5, 2 1 = = 5, 7, 9, 11, 13 8. a d 6, 8 1 = = 6, 14, 22, 30, 38 9. a d 15, 4 1 = − = − − − − 15, 11, 7, 3, 1 10. a d 11, 5 1 = − = − − − 11, 6, 1, 4, 9 11. a d 5, 2 1 = = − − − 5, 3, 1, 1, 3 12. a d 3, 4 1 = − = − − − − − − 3, 7, 11, 15, 19 In Exercises 13–18, determine the indicated term for the arithmetic sequence with the first term, a ,1 and common difference, d. 13. Determine a20 when a d 6, 3. 1 = = 63 14. Determine a22 when a d 7, 2. 1 = = − 35 − 15. Determine a30 when a d 18, 5. 1 = − = 127 16. Determine a27 when a d 16, 2. 1 = = − 36 − 17. Determine a150 when a d 85, 1. 1 = = − 64 − 18. Determine a175 when a d 20, 2. 1 = − = 328 In Exercises 19–24, write an expression for the general or nth term, a ,n of the arithmetic sequence. 19. … 1, 2, 3, 4, a n n = 20. … 2, 4, 6, 8, a n2 n = 21. … 1, 3, 5, 7, a n2 1 n = − 22. 1, 1, 3, 5, − … a n2 3 n = − 23. 5, 8, 11, 14, … a n3 2 n = + 24. − − … 7, 2, 3, 8, a n5 12 n = − In Exercises 25–30, determine the sum of the terms of the arithmetic sequence. The number of terms, n, is given. 25. … = n 1, 2, 3, 4, , 50; 50 1275 26. … = n 2, 4, 6, 8, , 100; 50 2550 27. … = n 1, 3, 5, 7, , 99; 50 2500 28. n 5, 9, 13, 17, , 101; 25 … = 1325 29. n 100, 90, 80, 70, , 60; 17 … − = 340 30. n 4, 11, 18, 25, , 193; 28 − − − − … − = −2758 In Exercises 31–36, write the first five terms of the geometric sequence with the first term, a ,1 and common ratio, r. 31. = = a r 2, 3 1 2, 6, 18, 54, 162 32. = = a r 4, 2 1 4, 8, 16, 32, 64 33. a r 5, 2 1 = = 5, 10, 20, 40, 80 34. a r 8, 1 1 2 = = 8, 4, 2, 1, 1 2 35. a r 3, 1 1 = − = − − − − 3, 3, 3, 3, 3 36. a r 6, 2 1 = − = − 6, 12, 24, 48, 96 − − − In Exercises 37– 42, determine the indicated term for the geometric sequence with the first term, a ,1 and common ratio, r. 37. Determine a5 when a r 6, 2 1 = = 96 38. Determine a8 when a r 1, 4 1 = = 16,384 39. Determine a20 when a r 2, 2 1 = − = 1,048,576 − 40. Determine a15 when a r 3, 2. 1 = = − 49,152 41. Determine a7 when a r 5, 3. 1 = − = 3645 − 42. Determine a7 when a r 3, 3. 1 = − = − 2187 − In Exercises 43– 48, write an expression for the general or nth term, a ,n for the geometric sequence. 43. … 4, 16, 64, 256, a 4 n n = 44. … 3, 12, 48, 192, a 3 4 n n 1 = ⋅ − 45. … 2, 1, , , 1 2 1 4 * 46. … 72, 12, 2, , 1 3 a 72 1 6 n n 1 = ⋅ ⎛ ⎝⎜ ⎞ ⎠⎟ − 47. − − − − … 16, 8, 4, 2, * 48. − − … 3, 6, 12, 24, * *See Instructor Answer Appendix SECTION 5.7
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