2025 Nov TZ3 P2 Q7

Andy and Jess each have \(\$5000\).
Andy invests the money in a new savings plan that will pay interest at the end of each month.
Andy will receive a fixed amount of interest each month. The amount received is \(0.315\%\) of the initial investment.

1. 1. Determine the amount of interest Andy will receive at the end of each month. Give this answer correct to two decimal places.

   2. Hence, determine the amount of interest Andy will receive each year.

   3. Write down an expression in the form \(5000 + pn\), where \(p \in \mathbb{Z}^+\), for the amount of money Andy will have in his savings plan at the end of \(n\) years.

   4. Hence, show that Andy will have \(\$5945\) in his savings plan at the end of \(5\) years.

      [4]

In this part, where appropriate, give all answers to the nearest dollar.
Jess invests her \(\$5000\) in a new account that pays \(3\%\) interest compounded annually.

2. 1. Determine the amount of money that will be in Jess's account at the end of \(5\) years.

   2. Hence, find the amount of interest Jess will receive in the \(5\) years.

      [3]

3. 1. Write an expression in the form \(5000 \times q^n\), where \(q \in \mathbb{R}^+\), for the amount of money that Jess will have in her account at the end of \(n\) years.

   2. Hence, determine the smallest number of complete years that it will take for Jess to have more money in her account than Andy has in his savings plan.

      [5]