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.

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]