1959 AMC 12 第 33 题

Question

调和数列是指其各项的倒数组成等差数列的数列。令

S_n

表示调和数列前

n

项之和；例如，

S_3

表示前三项之和。若某调和数列前三项为

3

、

4

、

6

，则：

A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression. Let

S_n

represent the sum of the first

n

terms of the harmonic progression; for example,

S_3

represents the sum of the first three terms. If the first three terms of a harmonic progression are

3,

4,

6,

then:

S_4=20

S_4=25

S_5=49

S_6=49

S_2=\dfrac12S_4

写出倒数

\frac{1}{3},\frac{1}{4},\frac{1}{6}

，并确定其公差

Write the reciprocals

\frac{1}{3},\frac{1}{4},\frac{1}{6}

and identify their common difference

将倒数组成的等差数列再延续一项

Continue the arithmetic progression of reciprocals one more term

各项的倒数开始为

\frac13,\quad\frac14,\quad\frac16\text{，}

公差为

-\frac{1}{12}

。下一个倒数是

\frac{1}{12}

，所以第四项为

12

。因此

S_4=3+4+6+12=25\text{。}

因此，正确答案是 B。

The reciprocals begin

\frac13,\quad\frac14,\quad\frac16,

with common difference

-\frac{1}{12}.

The next reciprocal is

\frac{1}{12},

so the fourth term is

12.

Therefore

S_4=3+4+6+12=25.

Thus, the correct answer is B.

其他年份的第 33 题