5.7 Arithmetic and Geometric Sequences 281 In Exercises 49–54, determine the sum of the first n terms of the geometric sequence for the values of a1 and r. 49. n a r 6, 3, 2 1 = = = 189 50. n a r 7, 1, 3 1 = = = 1093 51. n a r 6, 3, 4 1 = = − = 4095 − 52. n a r 9, 3, 5 1 = = − = 1,464,843 − 53. n a r 15, 2, 3 1 = = = 14,348,906 54. n a r 20, 4, 2 1 = = = 4,194,300 55. Determine the sum of the first 100 natural numbers. 5050 56. Determine the sum of the first 100 odd natural numbers. 10,000 57. Determine the sum of the first 100 even natural numbers. 10,100 58. Determine the sum of the first 50 multiples of 3. 3825 Problem Solving In Exercises 59– 68, when necessary, round your answer to the nearest hundredth. 59. Clock Strikes A clock strikes once at 1 o’clock, twice at 2 o’clock, three times at 3 o’clock, and so on. How many times does it strike over a 12-hr period? 78 times 60. A Bouncing Ball Each time a ball bounces, the height attained by the ball is 6 in. less than the previous height attained. If on the first bounce the ball reaches a height of 6 ft, determine the height attained on the eleventh bounce. 12 in. 61. Pendulum Movement Each swing of a pendulum is 3 in. shorter than the preceding swing. The first swing is 8 ft. a) Determine the length of the twelfth swing. 63 in. b) Determine the total distance traveled by the pendulum during the first 12 swings. 954 in. 62. Annual Pay Raises Jennifer is given a starting salary of $45,000 and promised a $1500 pay raise per year after each of the next 6 years. a) Determine her salary during her sixth year of work. $52,500 b) Determine the total salary she received over the 6 years. $292,500 63. Enrollment Increase The current enrollment at Linus College is 8000 students. If the enrollment increases by 8% per year, determine the enrollment 10 years later. 15,992 students 64. Samurai Sword Construction While making a traditional Japanese samurai sword, the master sword maker prepares the blade by heating a bar of iron until it is white hot. He then folds it over and pounds it smooth. Therefore, after each folding, the number of layers of steel is doubled. Assuming the sword maker starts with a bar of one layer and folds it 15 times, how many layers of steel will the finished sword contain? 32,768 layers 65. Salary Increase If your salary were to increase at a rate of 3% per year, determine your salary during your 15th year if your original salary was $45,000. (Hint: Use a 45,000, 1 = r 1.03, = and n 15. = ) $68,066.54 66. A Bouncing Ball When dropped, a ball rebounds to fourfifths of its original height. How high will the ball rebound after the fourth bounce if it is dropped from a height of 30 ft? 12.288 ft 67. France’s Population Growth In 2023 France’s population was about 68 million people. If France’s population is growing by about 0.20% per year, estimate France’s population in the year 2035. (Hint: Use a r 68, 1.002, 1 = = and n 12. = ) 69.51 million people 68. India’s Population Growth In 2023 India’s population was about 1.4 billion people. If India’s population is growing by about 0.81% per year, estimate India’s population in the year 2035. 1.53 billion people Challenge Problems/Group Activities 69. A geometric sequence has a 82 1 = and r ; 1 2 = determine s .6 161.4375 70. Sums of Interior Angles The sums of the interior angles of a triangle, a quadrilateral, a pentagon, and a hexagon are ° ° ° 180 , 360 , 540 , and ° 720 , respectively. Use this pattern to determine a formula for the general term, a ,n where an represents the sum of the interior angles of an n-sided polygon. a n n 180 360, 3 n = − ≥ 71. Divisibility by 6 Determine how many numbers between 7 and 1610 are divisible by 6. 267 72. Determine r and a1 for the geometric sequence with a 24 2 = and a 648. 5 = r a 3, 8 1 = = 73. Total Distance Traveled by a Bouncing Ball A ball is dropped from a height of 30 ft. On each bounce, it attains a height four-fifths of its original height (or of the previous bounce). Determine the total vertical distance traveled by the ball after it has completed its fifth bounce (has therefore hit the ground six times). 191.3568 ft Rudall30/Shutterstock Terry Oakley/Alamy Stock Photo
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