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

2014 AMC 12A Problem 7 — Solution

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

$1$

$\sqrt[7]{3}$

$\sqrt[8]{3}$

$\sqrt[9]{3}$

$\sqrt[10]{3}$

Writing the terms as powers of $3,$ they are $3^{1/2},$ $3^{1/3},$ $3^{1/6}.$ The common ratio is $\dfrac{3^{1/3}}{3^{1/2}}=3^{-1/6}.$

The fourth term is $3^{1/6}\cdot3^{-1/6}=3^{0}=1.$

Thus, the correct answer is A.