The first terms of an arithmetic sequence are $\frac{1}{{{{\log }_2}x}},{\text{ }}\frac{1}{{{{\log }_8}x}},{\text{ }}\frac{1}{{{{\log }_{32}}x}},{\text{ }}\frac{1}{{{{\log }_{128}}x}},{\text{ }} \ldots$

Find x if the sum of the first 20 terms of the sequence is equal to 100.

METHOD 1

$d = \frac{1}{{{{\log }_8}x}} - \frac{1}{{{{\log }_2}x}}$     (M1)

$= \frac{{{{\log }_2}8}}{{{{\log }_2}x}} - \frac{1}{{{{\log }_2}x}}$     (M1)

Note: Award this M1 for a correct change of base anywhere in the question.

$= \frac{2}{{{{\log }_2}x}}$     (A1)

$\frac{{20}}{2}\left( {2 \times \frac{1}{{{{\log }_2}x}} + 19 \times \frac{2}{{{{\log }_2}x}}} \right)$     M1

$= \frac{{400}}{{{{\log }_2}x}}$     (A1)

$100 = \frac{{400}}{{{{\log }_2}x}}$

${\log _2}x = 4 \Rightarrow x = {2^4} = 16$     A1

METHOD 2

${20^{{\text{th}}}}{\text{ term}} = \frac{1}{{{{\log }_{{2^{39}}}}x}}$     A1

$100 = \frac{{20}}{2}\left( {\frac{1}{{{{\log }_2}x}} + \frac{1}{{{{\log }_{{2^{39}}}}x}}} \right)$     M1

$100 = \frac{{20}}{2}\left( {\frac{1}{{{{\log }_2}x}} + \frac{{{{\log }_2}{2^{39}}}}{{{{\log }_2}x}}} \right)$     M1(A1)

Note: Award this M1 for a correct change of base anywhere in the question.

$100 = \frac{{400}}{{{{\log }_2}x}}$     (A1)

${\log _2}x = 4 \Rightarrow x = {2^4} = 16$     A1

METHOD 3

$\frac{1}{{{{\log }_2}x}} + \frac{1}{{{{\log }_8}x}} + \frac{1}{{{{\log }_{32}}x}} + \frac{1}{{{{\log }_{128}}x}} +  \ldots$

$\frac{1}{{{{\log }_2}x}} + \frac{{{{\log }_2}8}}{{{{\log }_2}x}} + \frac{{{{\log }_2}32}}{{{{\log }_2}x}} + \frac{{{{\log }_2}128}}{{{{\log }_2}x}} +  \ldots$     (M1)(A1)

Note: Award this M1 for a correct change of base anywhere in the question.

$= \frac{1}{{{{\log }_2}x}}(1 + 3 + 5 +  \ldots )$     A1

$= \frac{1}{{{{\log }_2}x}}\left( {\frac{{20}}{2}(2 + 38)} \right)$     (M1)(A1)

$100 = \frac{{400}}{{{{\log }_2}x}}$

${\log _2}x = 4 \Rightarrow x = {2^4} = 16$     A1

[6 marks]

There were plenty of good answers to this question. Those who realised they needed to make each log have the same base (and a great variety of bases were chosen) managed the question successfully.