5 A financial institution models the repayment of a loan to a client in the following way.

• An amount, \(\pounds C\), is loaned to the client at the start of the repayment period.
• The amount owed \(n\) years after the start of the repayment period is \(\pounds L _ { n }\), so that \(L _ { 0 } = C\).
• At the end of each year, interest of \(\alpha \% ( \alpha > 0 )\) of the amount owed at the start of that year is added to the amount owed.
• Immediately after interest has been added to the amount owed a repayment of \(\pounds R\) is made by the client.
• Once \(L _ { n }\) becomes negative the repayment is finished and the overpayment is refunded to the client.
  1. Show that during the repayment period, \(L _ { n + 1 } = a L _ { n } + b\), giving \(a\) and \(b\) in terms of \(\alpha\) and \(R\).
  2. Find the solution of the recurrence relation \(L _ { n + 1 } = a L _ { n } + b\) with \(L _ { 0 } = C\), giving your solution in terms of \(a , b , C\) and \(n\).
  3. Deduce from parts (a) and (b) that, for the repayment scheme to terminate, \(R > \frac { \alpha C } { 100 }\).

A client takes out a \(\pounds 30000\) loan at \(8 \%\) interest and agrees to repay \(\pounds 3000\) at the end of each year.

1. Use an algebraic method to find the number of years it will take for the loan to be repaid.
2. Taking into account the refund of overpayment, find the total amount that the client repays over the lifetime of the loan.