Question

数列是一个等差数列。求 $x$ 。 $\begin{gathered} \log_{12} 162, \ \log_{12} x, \ \log_{12} y, \\ \ \log_{12} z, \ \log_{12} 1250 \end{gathered}$

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 $x?$

$125\sqrt{3}$

$270$

$162\sqrt{5}$

$434$

$225\sqrt{6}$

答案：B

知识点：对数等比数列等差数列

难度评级：1800

解答：

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.

2013 AMC 12A 第 14 题

答案：B

难度评级：1800

解答：

其他年份的第 14 题