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

(Solution to Problem 12)
 
m (Problem 12)
 
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== Problem 12==
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== Problem ==
  
 
The first three terms of a geometric progression are <math>\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}</math>. Find the fourth term.
 
The first three terms of a geometric progression are <math>\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}</math>. Find the fourth term.
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\textbf{(C)}\ \sqrt[8]{2}\qquad
 
\textbf{(C)}\ \sqrt[8]{2}\qquad
 
\textbf{(D)}\ \sqrt[9]{2}\qquad
 
\textbf{(D)}\ \sqrt[9]{2}\qquad
\textbf{(E)}\ \sqrt[10]{2}</math>  
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\textbf{(E)}\ \sqrt[10]{2}</math>
  
 
==Solution==
 
==Solution==

Latest revision as of 16:44, 19 May 2018

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 (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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