Maths Olympiad Prep

Track / Stage 5 / 33 of 400 #633 of 1964

Problem 633

AIME late
Algebra Difficulty 5.1 Find the answer

1. If a+log⁡32017,a+log⁡92017,a+log⁡272017(a∈R)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

A number or a short expression. Spacing and $ signs are ignored.

Official solution

1. 13\frac{1}{3}.

Analysis: According to the problem, the common ratio q=a+log⁡92017a+log⁡32017=a+log⁡272017a+log⁡92017=log⁡92017−log⁡272017log⁡32017−log⁡92017q=\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}
=(12−13)log⁡32017(1−12)log⁡32017=13 =\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 and solution reproduced as published; topic, difficulty and ordering added by this site.