874 CHAPTER 12 Sequences; Induction; the Binomial Theorem In Problems 39–56, find each sum. 39. ( ) + + + + −n 1 3 5 2 1 40. + + + + n 2 4 6 2 41. ( ) + + + + + n 7 12 17 2 5 42. ( ) − + + + + −n 1 3 7 4 5 43. + + + + 2 4 6 70 44. + + + + 1 3 5 59 45. − − − + + 9 5 1 39 46. + + + + 2 5 8 41 47. + + + − 93 89 85 287 48. + − − − − 7 1 5 11 299 49. + + + + + 4 4.5 5 5.5 100 50. + + + + + + 8 8 1 4 8 1 2 8 3 4 9 50 51. ∑( ) − = n4 9 n 1 80 52. ∑( ) − = n 3 2 n 1 90 53. ∑( ) − = n 6 1 2 n 1 100 54. ∑( ) + = n 1 3 1 2 n 1 80 55. The sum of the first 120 terms of the sequence 14,16,18,20, . . . 56. The sum of the first 46 terms of the sequence − − − 2, 1, 4, 7, . . . Applications and Extensions 57. Find x so that + + x x 3,2 1, and +x5 2 are consecutive terms of an arithmetic sequence. 58. Find x so that + x x 2 ,3 2, and +x5 3 are consecutive terms of an arithmetic sequence. 59. How many terms must be added in an arithmetic sequence whose first term is 11 and whose common difference is 3 to obtain a sum of 1092? 60. How many terms must be added in an arithmetic sequence whose first term is 78 and whose common difference is −4 to obtain a sum of 702? 61. Drury Lane Theater The Drury Lane Theater has 25 seats in the first row and 30 rows in all. Each successive row contains one additional seat. How many seats are in the theater? 62. Seats in an Amphitheater An outdoor amphitheater has 35 seats in the first row, 37 in the second row, 39 in the third row, and so on. There are 27 rows altogether. How many can the amphitheater seat? 63. Football Stadium The corner section of a football stadium has 15 seats in the first row and 40 rows in all. Each successive row contains two additional seats. How many seats are in this section? 64. Constructing a Brick Staircase A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two fewer bricks than the prior step. (a) How many bricks are required for the top step? (b) How many bricks are required to build the staircase? 65. Salary If you take a job with a starting salary of $35,000 per year and a guaranteed raise of $1400 per year, how many years will it be before your aggregate salary exceeds $280,000? [Hint: Remember that your aggregate salary after 2 years is ( ) + + $35,000 $35,000 $1400 .] 66. Stadium Construction How many rows are in the corner section of a stadium containing 2040 seats if the first row has 10 seats and each successive row has 4 additional seats? 67. Creating a Mosaic A mosaic is designed in the shape of an equilateral triangle, 20 feet on each side. Each tile in the mosaic is in the shape of an equilateral triangle, 12 inches to a side. The tiles are to alternate in color as shown in the figure. How many tiles of each color will be required? 20' 20' 20' 68. Old Faithful Old Faithful is a geyser in Yellowstone National Park named for its regular eruption pattern. Past data indicates that the average time between eruptions is 1 h 35 min. (a) Suppose rangers log the first eruption on a given day at 12:57 am. Using = a 57, 1 write a prediction formula for the sequence of eruption times that day in terms of the number of minutes after midnight. (b) At what time of day (e.g., 7:15 am) is the 10th eruption expected to occur? (c) At what time of day is the last eruption expected to occur? 69. Cooling Air As a parcel of air rises (for example, as it is pushed over a mountain), it cools at the dry adiabatic lapse rate of ° 5.5 F per 1000 feet until it reaches its dew point. If the ground temperature is ° 67 F, write a formula for the sequence of temperatures, { } T , n of a parcel of air that has risen n thousand feet. What is the temperature of a parcel of air if it has risen 5000 feet? Source: National Aeronautics and Space Administration
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