2014 AMC 12A Problem 7

Problem 7 of 25EasierAlgebra

The first three terms of a geometric progression are

\sqrt{3},

\sqrt[3]{3},

and

\sqrt[6]{3}.

What is the fourth term?

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Solution

Writing the terms as powers of

3,

they are

3^{\frac{1}{2}},

3^{\frac{1}{3}},

3^{\frac{1}{6}}.

The common ratio is

\dfrac{3^{\frac{1}{3}}}{3^{\frac{1}{2}}}=3^{-\frac{1}{6}}.

The fourth term is

3^{\frac{1}{6}}\cdot3^{-\frac{1}{6}}=3^{0}=1.

Thus, the correct answer is A.

Concepts: geometric sequence · exponent

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.