Question $1$

$($a$)$ Paul breeds and sells rabbits for a living. He is planning for

his retirement by contributing to a retirement fund. He will invest

$500$ on each birthday from the age of $25$ to $64$ inclusive. That is

he will make $40$ contributions to the fund. The retirement fund

pays interest on the investments at the rate of $8\%$ per annum,

compounded anmually. How much money will be in Paul's fund

on his $65$ th birthday?

$($b$)$ On the day Paul's grandson Ollie was born,Paul deposited $5000$

into an account earning $3\%$ per annum compounded annually. On each

birthday after this he deposited $1000$ into the same account,making

his final deposit on Ollie's $17$th birthday.That is$,$ a total of $18$ deposits

were made. Let $A_n$ be the amount in the account on Ollie's nth birthday

after the deposit $n$ is made. Show that $A_3=8554\mathrm{.}54$

$($c$)$ Find the value of Ollie's account on his $17$th birthday,just after

the final deposit is made.