SECTION 12.1 Sequences 867 (a) Show that = u 1 1 and = u 1. 2 (b) Show that = + + + u u u . n n n 2 1 (c) Draw the conclusion that { } un is the Fibonacci sequence. 93. The Pascal Triangle Divide the triangular array shown (called the Pascal triangle) using diagonal lines as indicated. Find the sum of the numbers in each of these diagonal rows. Do you recognize this sequence? 1 6 15 20 15 6 1 1 5 10 10 5 1 1 4 6 4 1 1 3 3 1 1 2 1 1 1 1 94. Fibonacci Sequence Use the result of Problem 92 to do the following problems: (a) List the first 10 terms of the Fibonacci sequence. (b) Compute the ratio + u u n n 1 for the first 10 terms. (c) As n gets large, what number does the ratio approach? This number is referred to as the golden ratio. Rectangles whose sides are in this ratio were considered pleasing to the eye by the Greeks. For example, the facade of the Parthenon was constructed using the golden ratio. (d) Compute the ratio + u u n n 1 for the first 10 terms. (e) As n gets large, what number does the ratio approach? This number is also referred to as the conjugate golden ratio. This ratio is believed to have been used in the construction of the Great Pyramid in Egypt. The ratio equals the sum of the areas of the four face triangles divided by the total surface area of the Great Pyramid. 95. Approximating ( ) = f x ex In calculus, it can be shown that ∑ ( ) = = = ∞ f x e x k! x k k 0 We can approximate the value of ( ) = f x ex for any x using the following sum: ∑ ( ) = ≈ = f x e x k! x k n k 0 for some n. (a) Approximate ( ) f 1.3 with = n 4. (b) Approximate ( ) f 1.3 with = n 7. (c) Use a calculator to approximate ( ) f 1.3 . (d) Using trial and error along with a graphing utility’s SEQuence mode, determine the value of n required to approximate ( ) f 1.3 correct to eight decimal places. 96. Approximating ( ) = f x ex Refer to Problem 95. (a) Approximate ( ) − f 2.4 with = n 3. (b) Approximate ( ) − f 2.4 with = n 6. (c) Use a calculator to approximate ( ) − f 2.4 . (d) Using trial and error along with a graphing utility’s SEQuence mode, determine the value of n required to approximate ( ) − f 2.4 correct to eight decimal places. 89. Home Loan Bill and Laura borrowed $150,000 at 6% per annum compounded monthly for 30 years to purchase a home. Their monthly payment is determined to be $899.33. (a) Find a recursive formula for their balance after each monthly payment has been made. (b) Determine Bill and Laura’s balance after the first payment. (c) Using a graphing utility, create a table showing Bill and Laura’s balance after each monthly payment. (d) Using a graphing utility, determine when Bill and Laura’s balance will be below $140,000. (e) Using a graphing utility, determine when Bill and Laura will pay off the balance. (f) Determine Bill and Laura’s interest expense when the loan is paid. (g) Suppose that Bill and Laura decide to pay an additional $100 each month on their loan. Answer parts (a) to (f) under this scenario. (h) Is it worthwhile for Bill and Laura to pay the additional $100? Explain. 90. Home Loan Jodi and Jeff borrowed $120,000 at 6.5% per annum compounded monthly for 30 years to purchase a home. Their monthly payment is determined to be $758.48. (a) Find a recursive formula for their balance after each monthly payment has been made. (b) Determine Jodi and Jeff’s balance after the first payment. (c) Using a graphing utility, create a table showing Jodi and Jeff’s balance after each monthly payment. (d) Using a graphing utility, determine when Jodi and Jeff’s balance will be below $100,000. (e) Using a graphing utility, determine when Jodi and Jeff will pay off the balance. (f) Determine Jodi and Jeff’s interest expense when the loan is paid. (g) Suppose that Jodi and Jeff decide to pay an additional $100 each month on their loan. Answer parts (a) to (f) under this scenario. (h) Is it worthwhile for Jodi and Jeff to pay the additional $100? Explain. 91. Growth of a Rabbit Colony A colony of rabbits begins with one pair of mature rabbits, which will produce a pair of offspring (one male, one female) each month. Assume that all rabbits mature in 1 month and produce a pair of offspring (one male, one female) after 2 months. If no rabbits ever die, how many pairs of mature rabbits are there after 7 months? [Hint: The Fibonacci sequence models this colony. Do you see why?] 1 mature pair 1 mature pair 2 mature pairs 3 mature pairs 92. Fibonacci Sequence Let ( ) ( ) = + − − u 1 5 1 5 2 5 n n n n define the nth term of a sequence.
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