17M.2.SL.TZ2.S_5

pestleMathematicsAISLPaper 217M· sl-1-3-geometric-sequences-and-seriessource ↗

Consider a geometric sequence where the first term is 768 and the second term is 576.

Find the least value of $n$ such that the $n$ th term of the sequence is less than 7.

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt to find $r$ (M1)

eg $\,\,\,\,\,$ $\frac{{576}}{{768}},{\text{ }}\frac{{768}}{{576}},{\text{ }}0.75$

correct expression for ${u_n}$ (A1)

eg $\,\,\,\,\,$ $768{(0.75)^{n - 1}}$

EITHER (solving inequality)

valid approach (accept equation) (M1)

eg $\,\,\,\,\,$ ${u_n} < 7$

valid approach to find $n$ M1

eg $\,\,\,\,\,$ $768{(0.75)^{n - 1}} = 7,{\text{ }}n - 1 > {\log _{0.75}}\left( {\frac{7}{{768}}} \right)$ , sketch

correct value

eg $\,\,\,\,\,$ $n = 17.3301$ (A1)

$n = 18$ (must be an integer) A1 N2

OR (table of values)

valid approach (M1)

eg $\,\,\,\,\,$ ${u_n} > 7$ , one correct crossover value

both crossover values, ${u_{17}} = 7.69735$ and ${u_{18}} = 5.77301$ A2

$n = 18$ (must be an integer) A1 N2

OR (sketch of functions)

valid approach M1

eg $\,\,\,\,\,$ sketch of appropriate functions

valid approach (M1)

eg $\,\,\,\,\,$ finding intersections or roots (depending on function sketched)

correct value

eg $\,\,\,\,\,$ $n = 17.3301$ (A1)

$n = 18$ (must be an integer) A1 N2

[6 marks]

Examiners’ report

[N/A]