Question

数 $\log(a^3 b^7)$ 、 $\log(a^5 b^{12})$ 、 $\log(a^8 b^{15})$ 是一个等差数列的前三项，且该数列第 $12$ 项为 $\log(b^n)$ 。求 $n$ ？

The numbers $\log(a^3 b^7),$ $\log(a^5 b^{12}),$ and $\log(a^8 b^{15})$ are the first three terms of an arithmetic sequence, and the $12$ th term of the sequence is $\log(b^n).$ What is $n?$

$40$

$56$

$76$

$112$

$143$

答案：D

知识点：等差数列对数

难度评级：1800

解答：

三项分别为 $3\log a + 7\log b$ 、 $5\log a + 12\log b$ 、 $8\log a + 15\log b$ 。令相邻差相等，得 $\log a = 2\log b$ 。 $\begin{aligned} &2\log a + 5\log b \\ &= 3\log a + 3\log b, \end{aligned}$

于是第一项为 $(3 \cdot 2 + 7)\log b = 13\log b$ ，公差为 $(2 \cdot 2 + 5)\log b = 9\log b$ 。

第 $12$ 项为所以 $n = 112$ 。 $\begin{aligned} (13 + 11 \cdot 9)\log b &= 112\log b \\ &= \log(b^{112}), \end{aligned}$

所以正确答案是 D。

The three terms are $3\log a + 7\log b,$ $5\log a + 12\log b,$ and $8\log a + 15\log b.$ Setting the two consecutive differences equal, $\begin{aligned} &2\log a + 5\log b \\ &= 3\log a + 3\log b, \end{aligned}$ so $\log a = 2\log b.$

The first term is then $(3 \cdot 2 + 7)\log b = 13\log b,$ and the common difference is $(2 \cdot 2 + 5)\log b = 9\log b.$

The $12$ th term is $\begin{aligned} (13 + 11 \cdot 9)\log b &= 112\log b \\ &= \log(b^{112}), \end{aligned}$ so $n = 112.$

Thus, D is the correct answer.

2008 AMC 12A 第 16 题

答案：D

难度评级：1800

解答：

其他年份的第 16 题