1. A student takes out a loan for \(\pounds 1000\) from a bank.

The bank charges \(0.5 \%\) monthly interest on the amount of the loan yet to be repaid.
At the end of each month

• the interest is added to the loan
• the student then repays \(\pounds 50\)

Let \(U _ { n }\) be the amount of money owed \(n\) months after the loan was taken out.
The amount of money owed by the student is modelled by the recurrence relation $$U _ { n } = 1.005 U _ { n - 1 } - A \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$ where \(A\) is a constant.

1. 1. State the value of the constant \(A\).
   2. Explain, in the context of the problem, the value 1.005 Using the value of \(A\) found in part (a)(i),
2. solve the recurrence relation $$U _ { n } = 1.005 U _ { n - 1 } - A \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$
3. Hence determine, according to the model, the number of months it will take to completely repay the loan.