Question 1
Paper 2Hard6 marksTwo accounts each receive a deposit of $10 000. The principal is returned in full by either account, so only interest and fees are compared.
Account A pays simple interest at per year and has no fees.
The annual interest payments observed for account B are shown below.
| Year | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Interest payment ($) | 246 | 267 | 284 | 306 |
For a projection, an adviser models account B’s annual interest payments as an arithmetic sequence. The first term is the payment in year 1, and the common difference is the mean of the three consecutive changes shown in the table. Assume this model continues for years. Account B has a one-time opening fee of $200.
State the first term and common difference of the arithmetic model for account B.
Let be the total interest paid by account B during the first years. Show that
Hence determine the first whole number of years for which account B’s cumulative interest, after its opening fee, exceeds account A’s cumulative interest. Justify that this is the first such year.