N2012 P2 Q4

On 1 January 2001 Mrs A put $100 into a bank account, and on the first day of each subsequent month she put in $10 more than in the previous month. Thus on 1 February she put $110 into the account and on 1 March she put $120 into the account, and so on. The account pays no interest.

1. On what date did the value of Mrs A’s account first become greater than $5000?[5]

On 1 January 2001 Mr B put $100 into a savings account, and on the first day of each subsequent month he put another $100 into the account. The interest rate was 0.5% per month, so that on the last day of each month the amount in the account on that day was increased by 0.5%.

2. Use the formula for the sum of a geometric progression to find an expression for the value of Mr B’s account on the last day of the \(n\)th month (where January 2001 was the 1st month, February 2001 was the 2nd month, and so on). Hence find in which month the value of Mr B’s account first became greater than $5000.[5]
3. If Mr B wanted the value of his account to be $5000 on 2 December 2003, what interest rate per month, applied from January 2001, would achieve this?[3]