Use a recurrence relation to model the present value of an ordinary annuity, e.g. reducing balance loan or retirement pension with periodic payments where interest is calculated before the periodic payment is made.  𝐴𝑛+1 = 𝑟𝐴𝑛 − 𝑑 where 𝐴𝑛+1 is total amount at the beginning of the (𝑛 + 1)th period, 𝐴𝑛 is total amount at the beginning of the 𝑛th period, 𝑑 is periodic payment, and 𝑟 = 1 + 𝑖 where 𝑖 is interest rate per compounding period

A reducing balance loan for $\$18{,}500$ is subject to an interest rate of $7.2\%$ per annum, compounded monthly. The borrower makes monthly repayments of $\$340$. (a) Determine the value of $r$ in the recurrence relation $A_{n+1} = rA_n - d$, where $A_n$ is the amount owing at the beginning of month $n$. [1 mark] (b) Calculate the amount owing at the beginning of month 2. [1 mark] (c) Calculate the amount owing at the beginning of month 3. [1 mark]

A reducing balance loan of $\$28{,}000$ is issued at $7.2\%$ per annum, compounded monthly, with monthly repayments of $\$315$. (a) Write the recurrence relation that models the amount owing at the beginning of each month. (1 mark) (b) Use your recurrence relation to calculate the amount owing after two months, to the nearest cent. (2 marks)

A reducing balance loan of $18,500 is taken out with an interest rate of 6.6% per annum, compounded monthly. A fixed payment of $380 is made at the end of each month. Which recurrence relation models the loan balance \(A_n\) (in dollars) at the beginning of month \(n\)?

Marcus takes out a reducing balance loan of $18,500 to purchase a car. The loan has an interest rate of 7.2% p.a., compounded monthly, with monthly repayments of $345. (a) Determine the value of \(r\) for this loan. [1 mark] (b) Write a recurrence relation for the amount owing, \(A_n\), after \(n\) months. [1 mark] (c) Use the recurrence relation to calculate the amount Marcus will owe after 3 months, to the nearest cent. [2 marks]

A reducing balance loan of $18,000 is taken out at 7.2% p.a. compounding monthly. Monthly repayments of $325 are made at the end of each month. Use the recurrence relation \(A_{n+1} = rA_n - d\) to determine the amount owing at the beginning of the third month, to the nearest cent.