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]