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.