1961 AMC 12 第 12 题

Question

一个等比数列的前三项为

\sqrt2

、

\sqrt[3]2

、

\sqrt[6]2

。求第四项。

The first three terms of a geometric progression are

\sqrt2,

\sqrt[3]2,

\sqrt[6]2.

Find the fourth term.

1

\sqrt[7]2

\sqrt[8]2

\sqrt[9]2

\sqrt[10]2

将每个根式写成

2

的幂

Write each radical as a power of

2

检查相邻指数之间的差

Check the differences between the successive exponents

各项的指数为

\frac{1}{2}

、

\frac{1}{3}

、

\frac{1}{6}

。每一项都是前一项乘以

2^{-\frac{1}{6}}

得到的，所以后一项的指数为

\frac{1}{6}-\frac{1}{6}=0

。第四项是

2^0=1

。

所以，正确答案是 A。

The exponents are

\frac{1}{2},

\frac{1}{3},

and

\frac{1}{6}.

Each term is obtained by multiplying by

2^{-\frac{1}{6}},

so the next exponent is

\frac{1}{6}-\frac{1}{6}=0.

The fourth term is

2^0=1.

Thus, the correct answer is A.

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