Fixed monthly interest as an arithmetic model compared with annual compound interest over complete years

Priya and Marcus each have \(\$6000\) to save.

Priya puts her money into a savings plan that pays a fixed amount of interest at the end of every month. Each payment is \(0.2875\%\) of the amount she first invested.

1. 1. Find the amount of interest Priya receives at the end of each month, giving your answer correct to two decimal places.
   2. Hence, find the amount of interest Priya receives in one year.
   3. Write down an expression of the form \(6000+pn\), where \(p\in\mathbb{Z}^+\), for the amount in Priya's savings plan at the end of \(n\) years.
   4. Hence, show that Priya will have \(\$7242\) in her savings plan at the end of \(6\) years.[4]

In the rest of this question, give each amount of money to the nearest dollar where appropriate. Marcus puts his \(\$6000\) into an account that pays \(2.5\%\) interest per year, compounded annually.

2. 1. Find the amount in Marcus's account at the end of \(6\) years.
   2. Hence, find the total interest Marcus receives in these \(6\) years.[3]
3. 1. Write down an expression of the form \(6000\times q^n\), where \(q\in\mathbb{R}^+\), for the amount in Marcus's account at the end of \(n\) years.
   2. Hence, find the least number of complete years after which Marcus has more money in his account than Priya has in her savings plan.[5]