Date | May 2015 | Marks available | 3 | Reference code | 15M.1.sl.TZ2.9 |
Level | SL only | Paper | 1 | Time zone | TZ2 |
Command term | Find | Question number | 9 | Adapted from | N/A |
Question
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.
Markscheme
(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)