1. Sam borrows \(\pounds 10000\) from a bank to pay for an extension to his house.

The bank charges \(5 \%\) annual interest on the portion of the loan yet to be repaid. Immediately after the interest has been added at the end of each year and before the start of the next year, Sam pays the bank a fixed amount, \(\pounds F\). Given that \(\pounds A _ { n }\) (where \(A _ { n } \geqslant 0\) ) is the amount owed at the start of year \(n\),

1. write down an expression for \(A _ { n + 1 }\) in terms of \(A _ { n }\) and \(F\),
2. prove, by induction that, for \(n \geqslant 1\) $$A _ { n } = ( 10000 - 20 F ) 1.05 ^ { n - 1 } + 20 F$$
3. Find the smallest value of \(F\) for which Sam can repay all of the loan by the start of year 16 .