(Problem 4, continued) E. [1 pts] Recall that An = 20000 - 8000(1.015)" 3 i. [3 pts] Use algebra to find the exact expression for n so An = 0. ii. [1 pt] Use a calculator or other form of technology to report n to one decimal place. F. [8 pts] How much money will Jim have to pay in total to pay off the diamond-encrusted donut if he follows the repayment scheme described at the beginning of the problem? Hint: Go back and read how the repayments work carefully and what the formula derived for An gives. Note that there will only be a partial payment the last month, so Jim will not make a full payment during it.Problem 4: [30 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 are typically very,r 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 nances 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 rst two months is listed below. I At the start of Month 1, the balance has grown to 4000 x 1.015 : 4060. 0 The day before the end of the month, Jim pays $100 dollars. The balance is now $3960. I- At the start of Month 2, the previous balance of $3960 grows to 3960 x 1.015 = 4019.40 I The day before the end of the month, Jim pays $100 dollars. The balance is new $3919.40. The payment scheme is repeated until the the balance is eliminated. If the lmlanoe owed dining the last month is less than $100, then Jim will only pay the amolmt of the balance. A. [3 pts] Show that, to 2 decimal places, the balances at the start of Months 3 and 4 are $3818.19 and $3836.36, respectively. Let A\" denote the balance at the ad of Month n for each month where the balance is positive. To nd a formula for A\(Problem 4, continued) B. [6 pts] (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 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) AA-RN+1 1 _ R m the space below. N C. [5 pts] Derive the formula EA - R" : 3::0 D. [4 [1115] Apply the formula in Part C to the summation in your explicit formula from Part B. Then, do some algebra to Show that. _ 2mm 8000(1.[}15}\" A\" d