Use a recurrence relation to model a compound interest loan or investment.  𝐴𝑛+1 = 𝑟𝐴𝑛 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, and 𝑟 = 1 + 𝑖 where 𝑖 is interest rate per compounding period

Sarah invests \$18{,}500 into a term deposit account that compounds interest quarterly. The account's performance over the first year is shown in the table below. (a) Use the table to determine the quarterly interest rate \(i\) for this investment, expressing your answer as a percentage correct to two decimal places. (2 marks) (b) Write a recurrence relation in the form \(A_{n+1} = rA_n\) that models the value of Sarah's investment at the beginning of quarter \(n+1\), where \(A_0 = 18{,}500\). (1 mark) (c) Using your recurrence relation from part (b), calculate the value of Sarah's investment at the beginning of the third year (after 8 quarters). Give your answer to the nearest cent. (2 marks)

A startup investment of $\$8{,}500$ earns compound interest at $6.5\%$ per annum, compounded annually. Using the recurrence relation $A_{n+1} = rA_n$, which of the following represents the value of the investment after 5 years?

A town's current population of 12,400 is predicted to grow at an annual rate of 3.5%. (a) Write down the value of r in the recurrence relation A_{n+1} = rA_n that models this growth. (1 mark) (b) Using this model, predict the population at the beginning of year 6. Give your answer to the nearest whole number. (2 marks)