2020 TJC Promo Q12

A bank offers both an ordinary account and a savings account on 1st January 2020 Perlin puts \(\$100\) into an ordinary account. On the first day of each subsequent month she saves \(\$5\) more than in the previous month, so that she saves \(\$105\) on 1st February 2020, \(\$110\) on 1st March 2020, and so on. This account pays no interest.

1. On what date will the amount in Perlin's account first exceed \(\$5000\)?[5]

On 1st January 2020, Pauline puts \(\$100\) into a savings account, and on the first day of each subsequent month she puts another \(\$x\) into the account. The interest rate is \(0.5\%\) per month, so that on the last day of each month the amount in the account on that day is increased by \(0.5\%\).

2. Show that the expression for the amount in Pauline’s account on the last day of the \({{n}^{\mathrm{th}}}\) month (where January 2020 is the 1st month, February 2020 is the 2nd month, and so on) is given by
   \({{1.005}^{n}}\left( 200x+100 \right)-201x\).[3]
   1. If \(x=150\), find the number of complete months for the total amount in Pauline's account to first exceed \(\$5,000\)[2]
   2. Pauline wishes for the amount in her account to be at least \(\$5,000\) on 31st October 2021 in order to purchase a first-class flight to Japan for her year-end vacation. What is the smallest integer value of \(x\) would achieve this?[2]