Use a recurrence relation to model the future value of an ordinary annuity, e.g. compound interest investment 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

An annuity account has a balance of $5000 at the beginning of the first month. Each month, interest is calculated at 0.5% per month on the current balance, then a deposit of $300 is made. Which recurrence relation models the balance $A_n$ (in dollars) at the beginning of the nth month?

A retirement savings plan offers a compound interest investment with deposits made at the start of each quarter. An investor deposits $\$800$ at the beginning of each quarter into an account earning $6.8\%$ per annum, compounded quarterly. (a) Calculate the quarterly interest rate as a decimal. (1) (b) Write a recurrence relation for the total amount $A_n$ at the beginning of the $n$th quarter, where $A_0 = 800$. (2)

Maya opens a savings account with an initial deposit of \$3{,}500. The account earns 6.3% per annum with interest compounded quarterly. At the end of every quarter, Maya deposits an additional \$180 into the account. (a) Calculate the quarterly interest rate as a decimal. (1 mark) (b) Write a recurrence relation for the account balance, \(A_n\), where \(n\) is the number of quarters. (1 mark) (c) Use the recurrence relation to determine the balance after 2 quarters. Give your answer to the nearest cent. (2 marks)

Petra opens a superannuation account and makes a regular monthly contribution of $\$350$ into an investment fund. The fund earns interest at a rate of $4.8\%$ per annum, compounded monthly, and the interest is calculated at the beginning of each month before the contribution is deposited. (a) Calculate the monthly interest rate as a decimal. [1 mark] (b) Write a recurrence relation that models the amount in Petra's account, $A_n$, at the beginning of the $n$th month, where $A_0 = 0$. [1 mark] (c) Calculate the amount in the account at the beginning of the 4th month. [1 mark]

An investment account earns compound interest at $4.8\%$ per annum, compounded quarterly. The investor deposits $\$800$ at the beginning of each quarter. (a) Calculate the quarterly interest rate as a decimal. (1 mark) (b) Write a recurrence relation for the amount $A_n$ in the account at the beginning of the $n$th quarter. (1 mark) (c) Calculate the amount in the account at the beginning of the 5th quarter. (1 mark)

A savings account earns interest at a rate of $4.8\%$ per annum, compounded quarterly. Every three months, a deposit of $\$500$ is made at the beginning of each period, and interest is calculated on the existing balance before the deposit is added. Which recurrence relation correctly models the balance $A_n$ (in dollars) at the beginning of the $n$th quarter?