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1961 AHSME Problems/Problem 12 - Revision history
2024-03-29T12:20:08Z
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Rockmanex3: /* Problem 12 */
2018-05-19T21:44:20Z
<p><span dir="auto"><span class="autocomment">Problem 12</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 21:44, 19 May 2018</td>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>== Problem <del class="diffchange diffchange-inline">12</del>==</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>== Problem ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The first three terms of a geometric progression are <math>\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}</math>. Find the fourth term.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The first three terms of a geometric progression are <math>\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}</math>. Find the fourth term.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ \sqrt[8]{2}\qquad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(C)}\ \sqrt[8]{2}\qquad</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ \sqrt[9]{2}\qquad</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\textbf{(D)}\ \sqrt[9]{2}\qquad</div></td></tr>
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Rockmanex3
https://artofproblemsolving.com/wiki/index.php?title=1961_AHSME_Problems/Problem_12&diff=94634&oldid=prev
Rockmanex3: Solution to Problem 12
2018-05-19T21:43:29Z
<p>Solution to Problem 12</p>
<p><b>New page</b></p><div>== Problem 12==<br />
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The first three terms of a geometric progression are <math>\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}</math>. Find the fourth term.<br />
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<math>\textbf{(A)}\ 1\qquad<br />
\textbf{(B)}\ \sqrt[7]{2}\qquad<br />
\textbf{(C)}\ \sqrt[8]{2}\qquad<br />
\textbf{(D)}\ \sqrt[9]{2}\qquad<br />
\textbf{(E)}\ \sqrt[10]{2}</math> <br />
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==Solution==<br />
After rewriting the radicals as fractional exponents, the sequence is <math>2^{1/2}, 2^{1/3}, 2^{1/6}</math>.<br />
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The common ratio of the [[geometric sequence]] is <math>\frac{2^{1/3}}{2^{1/2}} = 2^{-1/6}</math>. Multiplying that by the third term results in <math>2^0</math>. It simplifies to <math>1</math>, so the answer is <math>\boxed{\textbf{(A)}}</math>.<br />
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==See Also==<br />
{{AHSME 40p box|year=1961|num-b=11|num-a=13}}<br />
<br />
[[Category:Introductory Algebra Problems]] <br />
<br />
{{MAA Notice}}</div>
Rockmanex3