Question

$ lo{{g}_{3}}2,\text{ }lo{{g}_{6}}2,\text{ }lo{{g}_{12}}2 $ are in

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

Answer: (C)

Given progression are
$ lo{{g}_{3}}2,lo{{g}_{6}}2,lo{{g}_{12}}2 $
Now, $ \frac{1}{{{\log }_{6}}2}-\frac{1}{{{\log }_{3}}2}={{\log }_{2}}6-{{\log }_{2}}3 $ $ ={{\log }_{2}}\left( \frac{6}{3} \right) $
$ ={{\log }_{2}}2=1 $
$ \frac{1}{{{\log }_{12}}2}-\frac{1}{{{\log }_{6}}2}={{\log }_{2}}12-{{\log }_{2}}6 $
$ ={{\log }_{2}}\left( \frac{12}{6} \right) $
$ ={{\log }_{2}}2=1 $
$ \because $ $ \frac{1}{{{\log }_{6}}2}-\frac{1}{{{\log }_{3}}2}=\frac{1}{{{\log }_{12}}2}-\frac{1}{{{\log }_{6}}2} $
$ \therefore $ $ {{\log }_{3}}2,{{\log }_{6}}2,{{\log }_{12}}2 $ are in HP.