Difference between revisions of "1961 AHSME Problems/Problem 12"
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| − | == Problem | + | == 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> | + | \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 ![$\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}$](http://latex.artofproblemsolving.com/2/c/4/2c4278a7094674dcd97ea8fb662cc33b1ecc3e9d.png)
. Find the fourth term.
![$\textbf{(A)}\ 1\qquad \textbf{(B)}\ \sqrt[7]{2}\qquad \textbf{(C)}\ \sqrt[8]{2}\qquad \textbf{(D)}\ \sqrt[9]{2}\qquad \textbf{(E)}\ \sqrt[10]{2}$](http://latex.artofproblemsolving.com/6/d/b/6db248c25e9e44259c4376bde53a3c11408c18bb.png)
Solution
After rewriting the radicals as fractional exponents, the sequence is 
.
The common ratio of the geometric sequence is 
. Multiplying that by the third term results in 
. It simplifies to 
, so the answer is 
.
See Also
| 1961 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 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. 
