Mr Lam saves $\$120$ on 1 January 2019. On the first day of each subsequent month, he saves \(\$10

Question

Mr Lam saves $\$120$ on 1 January 2019. On the first day of each subsequent month, he saves $\$10$ more than in the previous month so he saves $\$130$ on 1 February 2019, $\$140$ on 1 March 2019, and so on. On what date will he have first saved over $\$10\mathrm{}000$ in total?[4] Mr Lui has opened a special investment account that guarantees an interest rate of $8\%$ per annum, starting from the first year. He deposits $\$50\mathrm{}000$ into the account at the start of the first year and a fixed amount, $\$\,x$, at the beginning of each year from the second year onwards. The interest earned is based on the total amount fur the account at the end of that particular year and it is deposited into the account. For example, In Year One, at the beginning of the year there are $50000$ dollars in the account, and at the end of that year there are $\left( 1.08 \right)\left( 50000 \right)$ dollars. In Year Two. at the beginning of the year there are $\left[ (1.08)(50000)+x \right]$ dollars in the account, and at the end of that year there are $(1.08)\left[ (1.08)(50000)+x \right]$ dollars, which simplifies to $\left[ {{(1.08)}^{2}}(50000)+(1.08)x \right]$ dollars. Show that the total amount of money, in dollars, in the investment account at the end of the ${{n}^{\mathrm{th}}}$ year is $50000{{(1.08)}^{n}}+13.5x({{1.08}^{n-1}}-1)$.[3] If the yearly fixed amount that Mr Lui puts into the account from the second year onwards is $\$5\mathrm{}000$, find the total interest that he would earn at the end of the ${{10}^{\mathrm{th}}}$ year, correct to the nearest dollar.[2] Mr Lui is interested in opening another investment account that is similar to the special investment account but with an interest rate of $r\%$ per annum. If his deposit for the first year is still $\$50\mathrm{}000$, and the yearly fixed deposit thereafter is $\$5\mathrm{}000$, find the value of $r$ that would enable Mr Lui to have $\$300\mathrm{}000$ in the account after $10$ years.[3]

Accepted Answer

(a) 1 November 2021 (b)(ii) $80379\text{ dollars}$ (b)(iii) $r=15.1$

2020 CJC Promo Q12

2020 CJC Promo Q12

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