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(Problem 4, continued) B. (Formulas for A,) i. Give a recursive formula for A,. Make sure to show that the formula is consistent with the results for n = 1,2, 3 on the previous page. ii. Give an explicit formula for An that captures the pattern exhibited at the bottom of the previous page. The result should involve a sum, which you should write in summation notation, and an additional term. Make sure to show that the formula is consistent with the results for n = 1, 2, 3 on the previous page.

(Problem 4, continued) B. (Formulas for A,) i. Give a recursive formula for A,. Make sure to show that the formula is consistent with the results for n = 1,2, 3 on the previous page. ii. Give an explicit formula for An that captures the pattern exhibited at the bottom of the previous page. The result should involve a sum, which you should write in summation notation, and an additional term. Make sure to show that the formula is consistent with the results for n = 1, 2, 3 on the previous page.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(Problem 4, continued) B. (Formulas for An) i. Give a recursive formula for An. Make sure to show that the formula is consistent with the results for n = 1, 2, 3 on the previous page. ii. Give an explicit formula for A, that captures the pattern exhibited at the bottom of the previous page. The result should involve a sum, which you should write in summation notation, and an additional term. Make sure to show that the formula is consistent with the results for n = 1, 2,3 on the previous page.
Transcribed Image Text:(Problem 4, continued) B. (Formulas for An) i. Give a recursive formula for An. Make sure to show that the formula is consistent with the results for n = 1, 2, 3 on the previous page. ii. Give an explicit formula for A, that captures the pattern exhibited at the bottom of the previous page. The result should involve a sum, which you should write in summation notation, and an additional term. Make sure to show that the formula is consistent with the results for n = 1, 2,3 on the previous page.
Problem 4: An Application of Geometric Sums: Paying off credit card debt. Directions: Credit card debt is a major problem for many people. Interest rates for credit cards are typically very high, which makes paying down large debts quite expensive. This problem explores paying off credit card debt as an application of geometric series. To solidify his status as the "Calculus Sugar Daddy," Jim decides to buy a donut with diamonds for sprinkles. The bill comes to $4000, and Jim finances the purchase with a credit card whose annual interest rate is (a fairly typical) 18%. This is compounded monthly, meaning that at the start of every month, 1.5% interest is applied to the remaining balance. The repayment scheme of this purchase for the first two months is listed below. • At the start of Month 1, the balance has grown to 4000 x 1.015 = 4060. • The day before the end of the month, Jim pays $100 dollars. The balance is now $3960. • At the start of Month 2, the previous balance of $3960 grows to 3960 × 1.015 = 4019.40 • The day before the end of the month, Jim pays $100 dollars. The balance is now $3919.40. The payment scheme is repeated until the the balance is eliminated. If the balance owed during the last month is less than $100, then Jim will only pay the amount of the balance. Show that, to 2 decimal places, the balances at the start of Months 3 and 4 are $3878.19 and $3836.36, respectively. A. Let A, denote the balance at the end of Month n for each month where the balance is positive. To find a formula for An, we can do the following. For Month 1, note that A1 = 4000(1.015) – 100 For Month 2, note that A2 4000(1.015) – 100 (1.015) – 100 4000(1.015)² – 100 – 100(1.015) For Month 3, note that A3 |4000(1.015)2 – 100 – 100(1.015)|(1.015) – 100 = 4000(1.015)3 – 100 – 100(1.015) – 100(1.015)²
Transcribed Image Text:Problem 4: An Application of Geometric Sums: Paying off credit card debt. Directions: Credit card debt is a major problem for many people. Interest rates for credit cards are typically very high, which makes paying down large debts quite expensive. This problem explores paying off credit card debt as an application of geometric series. To solidify his status as the "Calculus Sugar Daddy," Jim decides to buy a donut with diamonds for sprinkles. The bill comes to $4000, and Jim finances the purchase with a credit card whose annual interest rate is (a fairly typical) 18%. This is compounded monthly, meaning that at the start of every month, 1.5% interest is applied to the remaining balance. The repayment scheme of this purchase for the first two months is listed below. • At the start of Month 1, the balance has grown to 4000 x 1.015 = 4060. • The day before the end of the month, Jim pays $100 dollars. The balance is now $3960. • At the start of Month 2, the previous balance of $3960 grows to 3960 × 1.015 = 4019.40 • The day before the end of the month, Jim pays $100 dollars. The balance is now $3919.40. The payment scheme is repeated until the the balance is eliminated. If the balance owed during the last month is less than $100, then Jim will only pay the amount of the balance. Show that, to 2 decimal places, the balances at the start of Months 3 and 4 are $3878.19 and $3836.36, respectively. A. Let A, denote the balance at the end of Month n for each month where the balance is positive. To find a formula for An, we can do the following. For Month 1, note that A1 = 4000(1.015) – 100 For Month 2, note that A2 4000(1.015) – 100 (1.015) – 100 4000(1.015)² – 100 – 100(1.015) For Month 3, note that A3 |4000(1.015)2 – 100 – 100(1.015)|(1.015) – 100 = 4000(1.015)3 – 100 – 100(1.015) – 100(1.015)²
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