When Jane was born on 1st January 2000, her grandmother decided to open a savings account for her that earns a fixed rate of 0.5% interest per month. A deposit of \(\$k\) is made at the start of every month, with the first deposit made on the day Jane was born. Interest is calculated on the total amount in the account at the end of every month, with the first interest added on the last day of January 2000.
Show that the total amount in Jane's account the day just before she turned 7 years old is \(\$201(1.005^{84}-1)k\).[3]
Determine the least integer value of \(k\) such that the total amount in Jane's account the day just before she turned 18 would be at least \(\$30\,000\).[2]
On 1st January 2018, she decided to withdraw \(\$20\,000\) from the savings account and put it in another account that earns a fixed rate of 0.6% interest per month. Every year on her birthday starting from 1st January 2019, she withdrew \(\$1800\) to buy herself a gift. Interest is calculated on the total amount in the account at the end of every month, with the first interest added on the last day of January 2018.
Determine the number of years she will be able to withdraw \(\$1800\) to buy herself a gift.[4]
The year that Jane found that there was less than \(\$1800\) to buy herself a gift, she decided to withdraw all the money and closed the account at the end of that year. How much would she get upon closing the account at the end of that year? Give your answer to the nearest cent.[2]
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Answer:(a)(i) \(201(1.005^{84}-1)k\) (a)(ii) \(k=78\) (b)(i) \(24\) years (b)(ii) \(\$859.07\)