9. An arithmetic series has first term \(a\) and common difference \(d\).
- Prove that the sum of the first \(n\) terms of the series is
$$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$
Sean repays a loan over a period of \(n\) months. His monthly repayments form an arithmetic sequence.
He repays \(\pounds 149\) in the first month, \(\pounds 147\) in the second month, \(\pounds 145\) in the third month, and so on. He makes his final repayment in the \(n\)th month, where \(n > 21\).
- Find the amount Sean repays in the 21st month.
Over the \(n\) months, he repays a total of \(\pounds 5000\).
- Form an equation in \(n\), and show that your equation may be written as
$$n ^ { 2 } - 150 n + 5000 = 0$$
- Solve the equation in part (c).
- State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.