One of the locations in the \(2016\) Olympic Games is an amphitheatre. The number of seats in the first row of the amphitheatre, \({u_1}\) , is \(240\). The number of seats in each subsequent row forms an arithmetic sequence. The number of seats in the sixth row, \({u_6}\) , is \(270\).

Calculate the value of the common difference, \(d\).

There are \(20\) rows in the amphitheatre.

Find the total number of seats in the amphitheatre.

Anisha visits the amphitheatre. She estimates that the amphitheatre has \(6500\) seats.

Calculate the percentage error in Anisha’s estimate.

\({\text{270}}\,{\text{ = }}\,{\text{240}}\,{\text{ + }}\,d\,(6 - 1)\)        (M1)

OR

\(d = \,\,\frac{{270 - 240}}{5}\)       (M1)

Note: Award (M1) for correct substitution into the arithmetic sequence formula.

\((d = )\,6\)       (A1) (C2)

\(\frac{{20}}{2}[2 \times 240 + 19 \times {\text{their }}d]\)       (M1)

Note: Award (M1) for correct substitution into sum of an arithmetic sequence.

OR

\({u_{20}} = 354\)

\({S_{20}} = \frac{{20}}{2}[240 + 354]\)      (M1)

Note: Award (M1) for correct substitution into sum of an arithmetic sequence.

OR
adding \(20\) terms consistent with their \(d\)       (M1)

 \( = 5940\)       (A1)(ft) (C2)

Note: Follow through from (a).

\(\left| {\frac{{6500 - 5940}}{{5940}}} \right| \times 100\)       (M1)

Note: Award (M1) for correct substitution into percentage error formula.

\( = 9.43\,(\% )\,\,\,(9.42760...)\)       (A1)(ft) (C2)

Note: Follow through from (b).

Question 6: Arithmetic sequence and series
This question was well attempted by the majority.
In part (a), a common error was calculating the difference as 5.

Part (b) was well attempted by the majority; with full follow-through being obtained.

In part (c) The incorrect denominator was the major error here.