| Date | May 2013 | Marks available | 6 | Reference code | 13M.1.hl.TZ1.8 |
| Level | HL only | Paper | 1 | Time zone | TZ1 |
| Command term | Find | Question number | 8 | Adapted from | N/A |
Question
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.
Markscheme
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]
Examiners report
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.
