Find \(r\).

Show that the sum of the infinite sequence is \(4{\log _2}x\).

Find \(d\), giving your answer as an integer.

Show that \({S_{12}} = 12{\log _2}x - 66\).

Given that \({S_{12}}\) is equal to half the sum of the infinite geometric sequence, find \(x\), giving your answer in the form \({2^p}\), where \(p \in \mathbb{Q}\).

evidence of dividing terms (in any order)     (M1)

eg\(\,\,\,\,\,\)\(\frac{{{\mu _2}}}{{{\mu _1}}},{\text{ }}\frac{{2{{\log }_2}x}}{{{{\log }_2}x}}\)

\(r = \frac{1}{2}\)    A1     N2

[2 marks]

correct substitution     (A1)

eg\(\,\,\,\,\,\)\(\frac{{2{{\log }_2}x}}{{1 - \frac{1}{2}}}\)

correct working     A1

eg\(\,\,\,\,\,\)\(\frac{{2{{\log }_2}x}}{{\frac{1}{2}}}\)

\({S_\infty } = 4{\log _2}x\)     AG     N0

[2 marks]

evidence of subtracting two terms (in any order)     (M1)

eg\(\,\,\,\,\,\)\({u_3} - {u_2},{\text{ }}{\log _2}x - {\log _2}\frac{x}{2}\)

correct application of the properties of logs     (A1)

eg\(\,\,\,\,\,\)\({\log _2}\left( {\frac{{\frac{x}{2}}}{x}} \right),{\text{ }}{\log _2}\left( {\frac{x}{2} \times \frac{1}{x}} \right),{\text{ }}({\log _2}x - {\log _2}2) - {\log _2}x\)

correct working     (A1)

eg\(\,\,\,\,\,\)\({\log _2}\frac{1}{2},{\text{ }} - {\log _2}2\)

\(d =  - 1\)    A1     N3

[4 marks]

correct substitution into the formula for the sum of an arithmetic sequence     (A1)

eg\(\,\,\,\,\,\)\(\frac{{12}}{2}\left( {2{{\log }_2}x + (12 - 1)( - 1)} \right)\)

correct working     A1

eg\(\,\,\,\,\,\)\(6(2{\log _2}x - 11),{\text{ }}\frac{{12}}{2}(2{\log _2}x - 11)\)

\(12{\log _2}x - 66\)    AG     N0

[2 marks]

correct equation     (A1)

eg\(\,\,\,\,\,\)\(12{\log _2}x - 66 = 2{\log _2}x\)

correct working     (A1)

eg\(\,\,\,\,\,\)\(10{\log _2}x = 66,{\text{ }}{\log _2}x = 6.6,{\text{ }}{2^{66}} = {x^{10}},{\text{ }}{\log _2}\left( {\frac{{{x^{12}}}}{{{x^2}}}} \right) = 66\)

\(x = {2^{6.6}}\) (accept \(p = \frac{{66}}{{10}}\))     A1     N2

[3 marks]