2013 AMC 12A 第 14 题

Question

数列

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

是一个等差数列。求

x

。

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}

对数的差相等，意味着

162, x, y, z, 1250

是等比数列

Equal differences of logs mean

162, x, y, z, 1250

is a geometric sequence

由

162r^4 = 1250

求公比

r

。

Find the common ratio

r

from

162r^4 = 1250

因为这些对数成等差数列，

162, x, y, z, 1250

成等比数列。它的公比

r

满足

162 r^4 = 1250

，所以

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

，且

r = \tfrac53

。

因此

x = 162\cdot\tfrac53 = 270

。

因此，正确答案是 B。

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,

so

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

and

r = \tfrac53.

Therefore

x = 162\cdot\tfrac53 = 270.

Thus, the correct answer is B.

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