Mrs Tan plans to start a business which requires a start-up capital of $\$ 700,000$. She decided to first save $\$ 200,000$ by depositing money every month into a savings plan. For the remaining $\$ 500,000$, she intends to take a loan from a finance company.
She deposited $\$ 3000$ into the savings plan in the first month and on the first day of each subsequent month, she deposited $\$ 100$ more than the previous month. Mrs Tan will continue depositing money into the savings plan until the total amount in her savings plan reaches $\$ 200,000$. It is given that this savings plan pays no interest.

(i) Find the month in which Mrs Tan’s monthly deposit will exceed $\$ 6,550$.

[2]

(ii) Find the number of months that it will take for Mrs Tan to save $\$ 200,000$ and hence find the amount that she would have deposited in the last month.

[4]

(iii) Show that the outstanding amount at the end of $n^{\text {th }}$ month, after the interest has been charged, is $A\left(1.003^n\right)-B x\left(1.003^n-1\right)$, where $A$ and $B$ are exact constants to be determined.

[3]

(iv) Find the amount of $x$, to 2 decimal places, if Mrs Tan wants to fully repay her loan in 8 years.

[2]

(v) Using the value of $x$ found in part (iv), calculate the total interest that the finance company will earn from Mrs Tan at the end of 8 years.

[2]