AMC12 2014 A

AMC12 2014 A · Q7

AMC12 2014 A · Q7. It mainly tests Exponents & radicals, Sequences & recursion (algebra).

The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term?

一个几何级数的头三项是$\sqrt 3$、$\sqrt[3]3$和$\sqrt[6]3$。第四项是多少？

(A) 1 1

(B) \sqrt[7]3 \sqrt[7]3

(D) \sqrt[9]3 \sqrt[9]3

(E) \sqrt[10]3\qquad \sqrt[10]3\qquad

Answer

Correct choice: (A)

正确答案：（A）

Solution

The terms are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$, which are equivalent to $3^{\frac{3}{6}}$, $3^{\frac{2}{6}}$, and $3^{\frac{1}{6}}$. So the next term will be $3^{\frac{0}{6}}=1$, so the answer is $\boxed{\textbf{(A)}}$.

各项为$\sqrt 3$、$\sqrt[3]3$和$\sqrt[6]3$，相当于$3^{\frac{3}{6}}$、$3^{\frac{2}{6}}$和$3^{\frac{1}{6}}$。下一项为$3^{\frac{0}{6}}=1$，故答案为$\boxed{\textbf{(A)}}$。

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