2014 AMC 12A Problems/Problem 7

Problem 7

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

$\textbf{(A) }1\qquad \textbf{(B) }\sqrt[7]3\qquad \textbf{(C) }\sqrt[8]3\qquad \textbf{(D) }\sqrt[9]3\qquad \textbf{(E) }\sqrt[10]3\qquad$

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 $\boxed{\textbf{(A)}}$ .

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2014 AMC 12A Problems/Problem 7

Problem 7

Solution