886 CHAPTER 12 Sequences; Induction; the Binomial Theorem For all but the center (largest) triangle, a triangle in the Koch snowflake is 1 9 the area of the next largest triangle in the fractal. Suppose the area of the largest triangle has area of 2 square meters. (a) Show that the area of the Koch snowflake is given by the series ( ) ( ) ( ) ( ) = + ⋅ + ⋅ + ⋅ + ⋅ + A 2 2 3 1 9 2 12 1 9 2 48 1 9 2 192 1 9 2 3 4 (b) Find the exact area of the Koch snowflake by finding the sum of the series. 101. A Rich Man’s Promise A rich man promises to give you $1000 on September 1. Each day thereafter he will give you 9 10 of what he gave you the previous day. What is the first date on which the amount you receive is less than 1¢? How much have you received when this happens? 102. Seating Revenue A special section in the end zone of a football stadium has 2 seats in the first row and 14 rows total. Each successive row has 2 seats more than the row before. In this particular section, the first seat is sold for 1 cent, and each following seat sells for 5% more than the previous seat. Find the total revenue generated if every seat in the section is sold. Round only the final answer, and state the final answer in dollars rounded to two decimal places. (JJC)† 103. Challenge Problem Suppose x, y, z are consecutive terms in a geometric sequence. If x y z 103 + + = and x y z 6901, 2 2 2 + + = find the value of y. [Hint: Let r be the common ratio so y xr = and z yr xr .2 = = ] 104. Challenge Problem Koch’s Snowflake The area inside the fractal known as the Koch snowflake can be described as the sum of the areas of infinitely many equilateral triangles, as pictured in the figure to the right. †Courtesy of the Joliet Junior College Mathematics Department Explaining Concepts 105. Critical Thinking You are interviewing for a job and receive two offers for a five-year contract: A: $40,000 to start, with guaranteed annual increases of 6% for the first 5 years B: $44,000 to start, with guaranteed annual increases of 3% for the first 5 years Which offer is better if your goal is to be making as much as possible after 5 years? Which is better if your goal is to make as much money as possible over the contract (5 years)? 106. Critical Thinking Which of the following choices, A or B, results in more money? A: To receive $1000 on day 1, $999 on day 2, $998 on day 3, with the process to end after 1000 days B: To receive $1 on day 1, $2 on day 2, $4 on day 3, for 19 days 107. Critical Thinking You have just signed a 7-year professional football league contract with a beginning salary of $2,000,000 per year. Management gives you the following options with regard to your salary over the 7 years. 1. A bonus of $100,000 each year 2. An annual increase of 4.5% per year beginning after 1 year 3. An annual increase of $95,000 per year beginning after 1 year Which option provides the most money over the 7-year period? Which the least? Which would you choose? Why? 108. Critical Thinking Suppose you were offered a job in which you would work 8 hours per day for 5 workdays per week for 1 month at hard manual labor.Your pay the first day would be 1 penny. On the second day your pay would be two pennies; the third day 4 pennies.Your pay would double on each successive workday. There are 22 workdays in the month. There will be no sick days. If you miss a day of work, there is no pay or pay increase. How much do you get paid if you work all 22 days? How much do you get paid for the 22nd workday? What risks do you run if you take this job offer? Would you take the job? 109. Can a sequence be both arithmetic and geometric? Give reasons for your answer. 110. Make up a geometric sequence. Give it to a friend and ask for its 20th term. 111. Make up two infinite geometric series, one that has a sum and one that does not. Give them to a friend and ask for the sum of each series. 112. Describe the similarities and differences between geometric sequences and exponential functions. Retain Your Knowledge Problems 113–122 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 113. Use the Change-of-Base Formula and a calculator to evaluate log 62. 7 Round the answer to three decimal places. 114. Find the unit vector in the same direction as = − v i j 8 15 . 115. Find the equation of the hyperbola with vertices at 2, 0 ( ) − and 2,0 , ( ) and a focus at 4, 0 . ( )
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