2024 RI Promo Q11

A financial institution, Future Investments Inc., has introduced a new investment scheme. The scheme pays a compound interest of \(5\%\) per annum at the end of the year, based on the amount in the account at the beginning of each year.

John and Sarah are both interested in this scheme.

1. John invests \(\$x\) at the start of the first year and a further \(\$x\) on the first day of each subsequent year. He chooses to leave the money in his account for the interest to accumulate.
   1. Write down the amount in John’s account, including the interest, at the end of the first year.[1]
   2. Show that John will have a total of \(21x\left( {{1.05}^{n}}-1 \right)\) in his account at the end of \(n\) years.[3]
   3. If John invests \(\$10000\) at the start of every year, find the number of years for the total in his account to first exceed \(\$500\,\,000\). Determine if this happens at the start or at the end of that year.[4]
2. Sarah invests \(\$6000\) at the start of the first year. On the first day of each subsequent year, she invests \(\$400\) more than the amount invested at the start of the previous year.
   1. Explain why the amount in Sarah’s account at the end of \(n\) years can be given by \(\sum\limits_{r=1}^{n}{\left( a+b\left( r-1 \right) \right)}{{\left( 1.05 \right)}^{n-\left( r-1 \right)}}\), and determine the values of the constants \(a\) and \(b\).[2]
   2. Hence determine the number of complete years for the total amount in her account to first exceed \(\$500\,\,000\).[2]