Only one of the following four sequences is arithmetic and only one of them is geometric.

     ${a_n} = 1,{\text{ }}2,{\text{ }}3,{\text{ }}5,{\text{ }} \ldots$

     ${b_n} = 1,{\text{ }}\frac{3}{2},{\text{ }}\frac{9}{4},{\text{ }}\frac{{27}}{8},{\text{ }} \ldots$

     ${c_n} = 1,{\text{ }}\frac{1}{2},{\text{ }}\frac{1}{3},{\text{ }}\frac{1}{4},{\text{ }} \ldots$

     ${d_n} = 1,{\text{ }}0.95,{\text{ }}0.90,{\text{ }}0.85,{\text{ }} \ldots$

State which sequence is

(i)     arithmetic;

(ii)     geometric.

For another geometric sequence ${e_n} =  - 6,{\text{ }} - 3,{\text{ }} - \frac{3}{2},{\text{ }} - \frac{3}{4},{\text{ }} \ldots$

write down the common ratio;

For another geometric sequence ${e_n} =  - 6,{\text{ }} - 3,{\text{ }} - \frac{3}{2},{\text{ }} - \frac{3}{4},{\text{ }} \ldots$

find the exact value of the tenth term. Give your answer as a fraction.

(i)     ${d_n}\;\;\;\;\;$OR$\;\;\;1,{\text{ }}0.95,{\text{ }}0.90,{\text{ }}0.85,{\text{ }} \ldots$     (A1)     (C1)

(ii)     ${b_n}\;\;\;$OR$\;\;\;1,{\text{ }}\frac{3}{2},{\text{ }}\frac{9}{4},{\text{ }}\frac{{27}}{8},{\text{ }} \ldots$     (A1)     (C1)

$\frac{1}{2}\;\;\;$OR$\;\;\;0.5$     (A1)     (C1)

Note: Accept ‘divide by 2’ for (A1).

$- 6{\left( {\frac{1}{2}} \right)^{10 - 1}}$     (M1)(A1)(ft)

Notes: Award (M1) for substitution in the GP ${n^{{\text{th}}}}$ term formula, (A1)(ft) for their correct substitution.

Follow through from their common ratio in part (b)(i).

OR

$\left( { - 6,{\text{ }} - 3,{\text{ }} - \frac{3}{2},{\text{ }} - \frac{3}{4},} \right) - \frac{3}{8},{\text{ }} - \frac{3}{{16}},{\text{ }} - \frac{3}{{32}},{\text{ }} - \frac{3}{{64}},{\text{ }} - \frac{3}{{128}}$     (M1)(A1)(ft)

Notes: Award (M1) for terms 5 and 6 correct (using their ratio).

Award (A1)(ft) for terms 7, 8 and 9 correct (using their ratio).

$- \frac{3}{{256}}\;\;\;\left( { - \frac{6}{{512}}} \right)$     (A1)(ft)     (C3)