Compound interest with yearly and quarterly compounding and a geometric series of savings deposits

Two savers each aim to build up a fund of \(\$30\,000\). Each of them has \(\$16\,000\) to invest at the start.

1. The first saver puts the \(\$16\,000\) into an account that pays interest at \(4.5\%\) per year, compounded yearly.
   1. Calculate the value of this investment after \(6\) years, giving your answer correct to the nearest hundred dollars.
   2. Find the least whole number of years after which the investment has reached the target.[5]
2. The second saver puts the \(\$16\,000\) into an account that pays interest at \(r\%\) per year, compounded quarterly, where \(r\) is given correct to \(2\) decimal places.
   Find the least value of \(r\) for which this investment reaches the target after \(8\) years.[3]
3. A third person, who has the same target, keeps savings in a cash box that earns no interest. A deposit is placed in the box at the start of each year, and each deposit after the first is two-thirds of the deposit made the year before.
   1. Show that, if the first deposit is \(\$9000\), the total in the box can never reach the target.
   2. Find the first deposit needed for the total in the box to reach the target when the fifth deposit is made. Give your answer correct to the nearest dollar.[8]