Find the value of $r$, the common ratio of the sequence.

Find the value of $n$ for which ${u_n} = 2$.

Find the sum of the first 30 terms of the sequence.

$\frac{{162}}{{486}}$$\,\,\,$OR$\,\,\,$$\frac{{54}}{{162}}$     (M1)

Note:     Award (M1) for dividing any ${u_{n + 1}}$ by ${u_n}$.

$= \frac{1}{3}{\text{ }}(0.333,{\text{ }}0.333333 \ldots )$     (A1)     (C2)

[2 marks]

$486{\left( {\frac{1}{3}} \right)^{n - 1}} = 2$     (M1)

Note:     Award (M1) for their correct substitution into geometric sequence formula.

$n = 6$     (A1)(ft)     (C2)

Note:     Follow through from part (a).

Award (A1)(A0) for ${u_6} = 2$ or ${u_6}$ with or without working.

[2 marks]

${S_{30}} = \frac{{486\left( {1 - {{\frac{1}{3}}^{30}}} \right)}}{{1 - \frac{1}{3}}}$     (M1)

Note:     Award (M1) for correct substitution into geometric series formula.

$= 729$     (A1)(ft)     (C2)

[2 marks]