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Question id: 20. This question is similar to one that appeared in an IB Studies paper in 2012. The use of a calculator is allowed.

Sequences

A Grecian amphitheatre was built in the form of a horseshoe and has 22 rows.

The number of seats in each row increase by a fixed amount, \(d\), compared to the number of seats in the previous row. The number of seats in the fifth row, \(u_5\), is 58, and the number of seats in the ninth row, \(u_{9}\), is 86. \(u_1\) represents the number of seats in the first row.

(a) Write an equation for \(u_5\) in terms of \(d\) and \(u_1\).

(b) Write an equation for \(u_{9}\) in terms of \(d\) and \(u_1\).

(c) Write down the value of \(d\);

(d) Write down the value of \(u_1\).

(e) Find the total number of seats in the amphitheatre.

Some time later, a second level was added to increase the amphitheatre’s capacity by another 2590 seats. Each row has five more seats than the previous row. The first row on this level has 82 seats.

(f) Find the number of rows on the second level of the amphitheatre.

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Dent

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