Mr Wong is considering investing money in a savings plan. One plane, $\mathrm{P}$, allows him to invest $\$100$ into the account on the first day of every month. At the end of each month, the total in the account is increased by $a\%$.

It is given that $\mathrm{a}=0.2$
1. Mr Wong invests $\$100$ on $1$ January $2016$. Write down how much this $\$100$ is worth at the end of $31$ December $2016$.
  [1]
2. Mr Wong invests $\$100$ on the first day of each of the $12$ months of $2016$. Find the total amount in the account at the end of $31$ December $2016$.
  [3]
3. Mr Wong continues to invest $\$100$ on the first day of each month. Find the month in which the total in the account will first exceed $\$3000$. Explain whether this occurs on the first or last day of the month.
  [5]

An alternative plan, $\mathrm{Q}$, also allows him to invest $\$100$ on the first day of every month. Each $\$100$ invested earns a fixed bonus of $\$\mathrm{b}$ at the end of every month for which it has been in the account. This is added to the account. The accumulated bonuses themselves do not earn any further bonus.

1. Find, in terms of $\mathrm{b}$, how much of $\$100$ invested on $1$ January $2016$ will be worth at the end of $31$ December $2016.$
  [1]
2. Mr Wong invests $\$100$ on the first day of each of the $24$ months in $2016$ and $2017$. Find the value of $\mathrm{b}$ such that the total value of all the investments, including bonuses, is worth $\$2800$ at the end of $31$ December $2017$.
  [3]

It is given instead that $\mathrm{a}=1$ for plan $\mathrm{P}$

Find the value of $\mathrm{b}$ for plan $\mathrm{Q}$ such that both plans give the same total value in the account at the end of the ${{60}^{\mathrm{th}}}$ month.
[3]

2018 TYS

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