# Difference between revisions of "1961 AHSME Problems/Problem 12"

## Problem

The first three terms of a geometric progression are $\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}$. Find the fourth term.

$\textbf{(A)}\ 1\qquad \textbf{(B)}\ \sqrt[7]{2}\qquad \textbf{(C)}\ \sqrt[8]{2}\qquad \textbf{(D)}\ \sqrt[9]{2}\qquad \textbf{(E)}\ \sqrt[10]{2}$

## Solution

After rewriting the radicals as fractional exponents, the sequence is $2^{1/2}, 2^{1/3}, 2^{1/6}$.

The common ratio of the geometric sequence is $\frac{2^{1/3}}{2^{1/2}} = 2^{-1/6}$. Multiplying that by the third term results in $2^0$. It simplifies to $1$, so the answer is $\boxed{\textbf{(A)}}$.

## See Also

 1961 AHSC (Problems • Answer Key • Resources) Preceded byProblem 11 Followed byProblem 13 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Invalid username
Login to AoPS