Problem 6 (26 pts]: 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 is 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 show off that he really is the "Calculus Sugar Daddy," Jim decides to by a donut that uses 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 x 1.015 = 4019.40 The day before the end of the month, Jim pays $100 dollars. The balance is now $3919.10. The payment scheme is repeated until the the balance is eliminated. A. [4 pts Show that, to 2 decimal places, the balances at the start of Months 3 and 4 are $3978.19 and $3936.36, respectively. Let Ar 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 Aj = 4000(1.015) - 100 For Month 2. note that A2 -100)(1.0 = 4000(1.015) - 100 (1.015) - 100 = 4000(1.015) - 100 - 100(1.015) 4000(1.015) - 100 - 100(1.015) (1.015) - 100 = 4000(1.015) - 100 - 100(1.015) - 100(1.015) For Month 3, note that A3 015)) (1 10 Print your name(s).nn here: B. 9 pts) The results for geometric series and pattern recognition can be used to find an explicit formula for Ar- i. 3 pts. Use the above results and pattern recognition to write down a formula for A. Make sure to show that your formula is consistent with the results for n=1,2,3 on the previous page. ii. 3 pts. Use the result shown in class) that ark G-N+ to show that 1- 100 + 100(1.015) + ... +100(1.015) 20000 (1 - (1.015)N+1) iii. 3 pts Use your formula from i, the result from ii, and some algebra to show that 20000 - 8000(1.015)" A = 11 Print your name(s).nn here: 20000 - 8000(1.015) C. 5 pts) Recall that An= i. [4 pts. Use algebra to find the exact expression for n so Ax=0 ii. (1 pt) Use a calculator or other form of technology to report n to one decimal place. D. (8 pts) How much money will Jim have to pay in total to pay off the diamond-encrusted donut? Hint: Note that there will only be a partial payment the last month