2026 RI P1 Q12

A seating specialist was tasked to install seats for a performance arena. In the curved section of the arena, he installed \(21\) seats in the front row facing the arena and for each subsequent row, there are \(2\) more seats than the row before it. The seating capacity of this curved section is \(1664\).

1. Find the total number of rows in this section of the arena, and the number of seats in the tenth row from the back of the section.[3]

At a performance held at this arena, the cost of a ticket for a front row seat at this curved section is $151. The cost of a ticket for a seat in every subsequent row decreases by \(2\%\) compared to the row before. The cost of tickets for seats in the same row are equal.

2. 1. Sam can only afford a ticket below $100 and he would like to sit as near to the front of the section as possible. Find the row nearest to the front for which he can purchase a ticket.[2]
   2. If \(\mathrm F(r)\) is the total cost of tickets for the seats in the \(r\)th row from the front of the section, write down \(\mathrm F(r)\) in terms of \(r\).[1]
   3. Find the total cost of tickets for all the seats in the first \(n\) rows from the front, in the form of
      \(\displaystyle143450(1-A^n)+B\sum_{r=1}^{n}\left(rA^{r-1}\right),\)
      where \(A\) and \(B\) are constants to be determined.
   4. Suppose a company can purchase tickets for this performance before public sales are open and they would like to purchase \(600\) tickets in this curved section with a budget of $52 000. Determine whether this budget is sufficient for this purchase.[3]