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]