The sequence log_12 162, log_12 x, log_12 y, log_12 z, log_12 1250 is an arithmetic progression. What is x?

2013 AMC 12A Problem 14

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14.

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?

125\sqrt{3}

270

162\sqrt{5}

434

225\sqrt{6}

Equal differences of logs mean

162, x, y, z, 1250

is a geometric sequence

Find the common ratio

r

from

162r^4 = 1250

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.