Question

A person borrows $1,950 on a bank credit card at a nominal rate of 18% per year, which is actually charged at a rate of 1.5% per month. (a) What is the annual percentage yield (APY) for the card? (Round your answer to one decimal place.) 19.56 % (b) Assume that the person does not place any additional charges on the card and pays the bank $150 each month to pay off the loan. Let B, be the balance owed on the card after n months. Find an explicit formula for B, (Enter a mathematical expression.) (c) How many months will be required to pay off the debt? (Round your answer up to the nearest whole number.) x months (d) What is the total amount of money the person will have paid for the loan?

Prove the following statement by mathematical induction. For every integer n > 0,2^n <(n + 2)!. Proof (by mathematical induction): Let P(n) be the inequality 2^n < (n + 2)!. We will show that P(n) is true for every integer n ≥0. (a) Show that P(0) is true. (For each answer, enter a mathematical expression.) 1. Before simplifying, the left-hand side of P(0) is 2. Before simplifying, the right-hand side of P(O) is The fact that the statement is true can be deduced from that fact that 2° = 1.

Answer 1

The annual percentage yield (APY) for the card is approximately 19.56%. The balance owed on the card after n months can be found using a formula. The number of months required to pay off the debt can be determined by solving an equation.

Step-by-step explanation:

(a) To calculate the annual percentage yield (APY), we can use the formula:

APY = (1 + r/m)^m - 1

Where r is the nominal rate (0.18), and m is the number of compounding periods in a year (12 for monthly compounding). Plugging in these values, we get:

APY = (1 + 0.18/12)^12 - 1

APY ≈ 0.1956, which is equivalent to 19.56% (rounded to one decimal place).

(b) The balance owed on the card after n months can be found using the formula:

B(n) = P(1 + r)^n - CP((1+r)^n - 1)/r

Where P is the initial principal (1950), r is the monthly interest rate (0.015), C is the monthly payment (150), and n is the number of months.

(c) To find the number of months required to pay off the debt, we need to solve the equation B(n) = 0 for n.

(d) The total amount of money the person will have paid for the loan can be calculated by multiplying the monthly payment (150) by the number of months required to pay off the debt (found in part c).