2013 AMC 12A Problem 14

Problem 14 of 25IntermediateAlgebra

The sequence

\begin{gathered} \log_{12} 162, \ \log_{12} x, \ \log_{12} y, \\ \ \log_{12} z, \ \log_{12} 1250 \end{gathered}

is an arithmetic progression. What is

x?

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Solution

Because the logarithms are in arithmetic progression,

162, x, y, z, 1250

is a geometric sequence. Its common ratio

r

satisfies

162 r^4 = 1250,

r^4 = \tfrac{625}{81}

and

r = \tfrac53.

Therefore

x = 162\cdot\tfrac53 = 270.

Thus, the correct answer is B.

Tagged: logarithm · geometric sequence · arithmetic sequence