866 CHAPTER 12 Sequences; Induction; the Binomial Theorem 65. ( ) ( ) − + − + + − 1 1 3 1 9 1 27 1 1 3 6 6 66. ( ) ( ) − + − + − 2 3 4 9 8 27 1 2 3 12 11 67. + + + + n 3 3 2 3 3 3n 2 3 68. + + + + e e e n e 1 2 3 n 2 3 69. ( ) ( ) ( ) + + + + + + + a a d a d a nd 2 70. + + + + − a ar ar arn 2 1 In Problems 71–82, find the sum of each sequence. 71. ∑ = 5 k 1 40 72. ∑ = 8 k 1 50 73. ∑ = k k 1 40 74. ∑( ) − = k k 1 24 75. ∑( ) + = k5 3 k 1 20 76. ∑( ) − = k3 7 k 1 26 77. ∑( ) + = k 4 k 1 16 2 78. ∑( ) − = k 4 k 0 14 2 79. ∑( ) = k2 k 10 60 80. ∑( ) − = k3 k 8 40 81. ∑ = k k 5 20 3 82. ∑ = k k 4 24 3 Applications and Extensions 83. Trout Population A pond currently has 2000 trout in it. A fish hatchery decides to add an additional 20 trout each month. In addition, it is known that the trout population is growing at a rate of 3% per month.The size of the population after n months is given by the recursively defined sequence = = + − p p p 2000 1.03 20 n n 0 1 (a) How many trout are in the pond at the end of the second month? That is, what is p ?2 (b) Using a graphing utility, determine how long it will be before the trout population reaches 5000. 84. Environmental Control The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of pollutants as a result of industrial waste and that 10% of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in 15 tons of new pollutant entering the lake each year. The amount of pollutant in the lake after n years is given by the recursively defined sequence = = + − p p p 250 0.9 15 n n 0 1 (a) Determine the amount of pollutant in the lake at the end of the second year. That is, determine p .2 (b) Using a graphing utility, provide pollutant amounts for the next 20 years. (c) What is the equilibrium level of pollution in Maple Lake? That is, what is →∞ p lim ? n n 85. Roth IRA On January 1, Liam deposits $1500 into a Roth individual retirement account (IRA) and decides to deposit an additional $750 at the end of each quarter into the account. (a) Find a recursive formula that represents Liam’s balance at the end of each quarter if the rate of return is assumed to be 5% per annum compounded quarterly. (b) Use a graphing utility to determine how long it will be before the value of the account exceeds $150,000. (c) What will be the value of the account in 30 years, when Liam retires? 86. Education Savings Account On January 1, Aubrey’s parents deposit $4000 in an education savings account and decide to place an additional $75 into the account at the end of each month. (a) Find a recursive formula that represents the balance at the end of each month if the rate of return is assumed to be 1.5% per annum compounded monthly. (b) Use a graphing utility to determine how long it will be before the value of the account exceeds $10,000. (c) What will be the value of the account in 16 years when Aubrey goes to college? 87. Credit Card Debt John has a balance of $3000 on his credit card that charges 1% interest per month on any unpaid balance from the previous month. John can afford to pay $100 toward the balance each month. His balance each month after making a $100 payment is given by the recursively defined sequence = = − − B B B $3000 1.01 100 n n 0 1 (a) Determine John’s balance after making the first payment. That is, determine B .1 (b) Using a graphing utility, determine when John’s balance will be below $2000. How many payments of $100 have been made? (c) Using a graphing utility, determine when John will pay off the balance. What is the total of all the payments? (d) What was John’s interest expense? 88. Car Loans Ivan bought a car by taking out a loan for $18,500 at 0.5% interest per month. Ivan’s normal monthly payment is $434.47 per month, but he decides that he can afford to pay $100 extra toward the balance each month. His balance each month is given by the recursively defined sequence = = − − B B B $18,500 1.005 534.47 n n 0 1 (a) Determine Ivan’s balance after making the first payment. That is, determine B .1 (b) Using a graphing utility, determine when Ivan’s balance will be below $10,000. How many payments of $534.47 have been made? (c) Using a graphing utility, determine when Ivan will pay off the balance. What is the total of all the payments? (d) What was Ivan’s interest expense?
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