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Problem 658

AIME late

Algebra Difficulty 5.1 Find the answer

If $a+\log _{3} 2017, a+\log _{9} 2017, a+\log _{27} 2017(a \in R)$ form a geometric sequence, then its common ratio is $\qquad$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

1. $\frac{1}{3}$ .

Analysis: According to the problem, the common ratio $q=\frac{a+\log _{9} 2017}{a+\log _{3} 2017}=\frac{a+\log _{27} 2017}{a+\log _{9} 2017}=\frac{\log _{9} 2017-\log _{27} 2017}{\log _{3} 2017-\log _{9} 2017}$
$=\frac{\left(\frac{1}{2}-\frac{1}{3}\right) \log _{3} 2017}{\left(1-\frac{1}{2}\right) \log _{3} 2017}=\frac{1}{3}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.