2014 RI Promo Q6

The annual wage of a certain occupation offered by Company \(P\) begins with an initial amount of \(\$9500\) and increases by \(\$400\) every year for \(14\) years till it reaches \(\$15100\) and remains constant at \(\$15100\) thereafter.

Let \(p(n) = a_1 + a_2 + a_3 + ... + a_n\), where \(a_i\) denotes the annual wage in the \(i^{\mathrm{th}}\) year offered by Company \(P\) for \(i = 1, 2, 3, \dots, n\).

1. Show that \(p(n) = \begin{cases} 200n^2 + 9300n & \text{for}\hspace{0.5em} n \le 15, \\ 15100n + A & \text{for}\hspace{0.5em} n > 15, \end{cases}\)
   where \(A\) is a constant to be determined.[4]
2. The annual wage of the same occupation offered by Company \(Q\) begins with the same initial amount of \(\$9500\) but it increases at a fixed rate of \(r \, \%\) every year.
   Sam has been offered a job for the same occupation at both Company \(P\) and \(Q\). Find the value of \(r\), correct to \(2\) decimal places, such that the total annual wages earned by Sam if he works for Company \(P\) for \(15\) years is equivalent to the total annual wages earned if he works for Company \(Q\) for \(12\) years.[4]