https://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_30&feed=atom&action=history
1988 AHSME Problems/Problem 30 - Revision history
2024-03-28T09:18:58Z
Revision history for this page on the wiki
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https://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_30&diff=92426&oldid=prev
Hapaxoromenon: Added the solution for the correct problem
2018-02-27T19:16:22Z
<p>Added the solution for the correct problem</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:16, 27 February 2018</td>
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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;"></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>==Solution==</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>==Solution==</div></td></tr>
<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><del class="diffchange diffchange-inline">Apply one of </del>the <del class="diffchange diffchange-inline">standard formulae for </del>the <del class="diffchange diffchange-inline">gradient of </del>the <del class="diffchange diffchange-inline">line of best fit</del>, <del class="diffchange diffchange-inline">e</del>.<del class="diffchange diffchange-inline">g</del>. <math><del class="diffchange diffchange-inline">\frac</del>{<del class="diffchange diffchange-inline">\frac</del>{<del class="diffchange diffchange-inline">\sum </del>{<del class="diffchange diffchange-inline">x_i y_i</del>}}{n} <del class="diffchange diffchange-inline">- \bar</del>{<del class="diffchange diffchange-inline">x</del>} <del class="diffchange diffchange-inline">\bar</del>{<del class="diffchange diffchange-inline">y</del>}}{\<del class="diffchange diffchange-inline">frac</del>{\<del class="diffchange diffchange-inline">sum </del>{<del class="diffchange diffchange-inline">x_</del>{<del class="diffchange diffchange-inline">i</del>}^2}}{n} - \<del class="diffchange diffchange-inline">bar</del>{<del class="diffchange diffchange-inline">x</del>}<del class="diffchange diffchange-inline">^2</del>}</math>, and <del class="diffchange diffchange-inline">substitute </del>in the <del class="diffchange diffchange-inline">given condition </del><math><del class="diffchange diffchange-inline">x_3 - x_2 = x_2 - x_1</del></math>. <del class="diffchange diffchange-inline">The answer is </del><math>\<del class="diffchange diffchange-inline">boxed{</del>\<del class="diffchange diffchange-inline">text</del>{<del class="diffchange diffchange-inline">A}</del>}</math>.</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><ins class="diffchange diffchange-inline">Note that <math>x_{0} = 0</math> gives </ins>the <ins class="diffchange diffchange-inline">constant sequence <math>0, 0, ...</math>, since <math>f(0) = 4 \cdot 0 - 0^2 = 0</math>. Because <math>f(4)=0, x_{0} = 4</math> gives </ins>the <ins class="diffchange diffchange-inline">sequence <math>4, 0, 0, ...</math> with two different values. Similarly, <math>f(2) = 4</math>, so <math>x_{0} = 2</math> gives </ins>the <ins class="diffchange diffchange-inline">sequence <math>2, 4, 0, 0</ins>, ..<ins class="diffchange diffchange-inline">.</math> with three values. In general, if <math>x_{0} = a_{n}</ins><<ins class="diffchange diffchange-inline">/</ins>math> <ins class="diffchange diffchange-inline">gives the sequence <math>a_{n}, a_{n-1}, ... , a_{2}, a_{1}, ...</math> with <math>n</math> different values, and <math>f(a_</ins>{<ins class="diffchange diffchange-inline">n+1}) = a_</ins>{<ins class="diffchange diffchange-inline">n}</math>, then <math>x_</ins>{<ins class="diffchange diffchange-inline">0</ins>} <ins class="diffchange diffchange-inline">= a_{n+1</ins>}<ins class="diffchange diffchange-inline"></math> gives a sequence with <math>n+1</math> different values. (It is easy to see that we could not have <math>a_</ins>{n<ins class="diffchange diffchange-inline">+1</ins>} <ins class="diffchange diffchange-inline">= a_</ins>{<ins class="diffchange diffchange-inline">i</ins>}<ins class="diffchange diffchange-inline"></math> for some <math>i < n + 1</math>.) Thus, it follows by induction that there is a sequence with <math>n</math> distinct values for every positive integer <math>n</math>, as long as we can verify that there is always a real number <math>a_</ins>{<ins class="diffchange diffchange-inline">n+1</ins>}<ins class="diffchange diffchange-inline"></math> such that <math>f(a_{n+1</ins>}<ins class="diffchange diffchange-inline">) = a_</ins>{<ins class="diffchange diffchange-inline">n}</math>. This makes the answer <math></ins>\<ins class="diffchange diffchange-inline">boxed</ins>{\<ins class="diffchange diffchange-inline">text{E}}</math>. The verification alluded to above, which completes the proof, follows from the quadratic formula: the solutions to <math>f(a_{n+1}) = 4a_</ins>{<ins class="diffchange diffchange-inline">n+1} - a_</ins>{<ins class="diffchange diffchange-inline">n+1</ins>}^<ins class="diffchange diffchange-inline">{</ins>2} <ins class="diffchange diffchange-inline">= a_{n</ins>}<ins class="diffchange diffchange-inline"></math> are <math>a_</ins>{n<ins class="diffchange diffchange-inline">+1</ins>} <ins class="diffchange diffchange-inline">= 2 \pm \sqrt{4 </ins>- <ins class="diffchange diffchange-inline">a_{n}}</math>. Hence if <math>0 </ins>\<ins class="diffchange diffchange-inline">leq a_</ins>{<ins class="diffchange diffchange-inline">n</ins>} <ins class="diffchange diffchange-inline">\leq 4</math>, then <math>a_{n+1</ins>}</math> <ins class="diffchange diffchange-inline">is real, since the part under the square root is non-negative</ins>, and in <ins class="diffchange diffchange-inline">fact <math>0 \leq a_{n+1} \leq 4</math>, since <math>4-a_{n}</math> will be between <math>0</math> and <math>4</math>, so </ins>the <ins class="diffchange diffchange-inline">square root will be between <math>0</math> and <math>2</math>, and <math>2 \pm</math> something between </ins><math><ins class="diffchange diffchange-inline">0</math> and <math>2</math> gives something between <math>0</math> and <math>4</ins></math>. <ins class="diffchange diffchange-inline">Finally, since </ins><math><ins class="diffchange diffchange-inline">a_{1} = 0 </ins>\<ins class="diffchange diffchange-inline">leq 4</math>, it follows by induction that all terms satisfy <math>0 </ins>\<ins class="diffchange diffchange-inline">leq a_</ins>{<ins class="diffchange diffchange-inline">n</ins>} <ins class="diffchange diffchange-inline">\leq 4</ins></math><ins class="diffchange diffchange-inline">; in particular, they are all real</ins>.</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>== See also ==</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>== See also ==</div></td></tr>
</table>
Hapaxoromenon
https://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_30&diff=92424&oldid=prev
Hapaxoromenon: Added a solution with explanation
2018-02-27T19:02:52Z
<p>Added a solution with explanation</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 19:02, 27 February 2018</td>
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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;"></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>==Solution==</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>==Solution==</div></td></tr>
<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> </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><ins class="diffchange diffchange-inline">Apply one of the standard formulae for the gradient of the line of best fit, e.g. <math>\frac{\frac{\sum {x_i y_i}}{n} - \bar{x} \bar{y}}{\frac{\sum {x_{i}^2}}{n} - \bar{x}^2}</math>, and substitute in the given condition <math>x_3 - x_2 = x_2 - x_1</math>. The answer is <math>\boxed{\text{A}}</math>.</ins></div></td></tr>
<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> </div></td><td colspan="2"> </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>== See also ==</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>== See also ==</div></td></tr>
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Hapaxoromenon
https://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_30&diff=78041&oldid=prev
Quantummech: /* Problem */
2016-04-14T09:56:50Z
<p><span dir="auto"><span class="autocomment">Problem</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 09:56, 14 April 2016</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6" >Line 6:</td>
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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><math>\textbf{(A)}\ \text{0}\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><math>\textbf{(A)}\ \text{0}\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{(B)}\ \text{1 or 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{(B)}\ \text{1 or 2}\qquad</div></td></tr>
<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>\textbf{(C)}\ \text{3, 4, 5 or 6}\qquad<del class="diffchange diffchange-inline">\\</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>\textbf{(C)}\ \text{3, 4, 5 or 6}\qquad</div></td></tr>
<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>\textbf{(D)}\ \text{more than 6 but finitely many}\qquad<del class="diffchange diffchange-inline">\\</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>\textbf{(D)}\ \text{more than 6 but finitely many}\qquad</div></td></tr>
<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>\textbf{(E) }\infty</math>  </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>\textbf{(E) }\infty</math></div></td></tr>
<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> </div></td><td colspan="2"> </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>==Solution==</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>==Solution==</div></td></tr>
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Quantummech
https://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_30&diff=65666&oldid=prev
Timneh: /* See also */
2014-10-23T12:29:11Z
<p><span dir="auto"><span class="autocomment">See also</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 12:29, 23 October 2014</td>
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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>== See also ==</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>== See also ==</div></td></tr>
<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>{{AHSME box|year=1988|num-b=<del class="diffchange diffchange-inline">28</del>|after=Last Question}}   </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>{{AHSME box|year=1988|num-b=<ins class="diffchange diffchange-inline">29</ins>|after=Last Question}}   </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>[[Category: Intermediate Algebra Problems]]</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>[[Category: Intermediate Algebra Problems]]</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>{{MAA Notice}}</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>{{MAA Notice}}</div></td></tr>
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Timneh
https://artofproblemsolving.com/wiki/index.php?title=1988_AHSME_Problems/Problem_30&diff=65601&oldid=prev
Timneh: Created page with "==Problem== Let <math>f(x) = 4x - x^{2}</math>. Give <math>x_{0}</math>, consider the sequence defined by <math>x_{n} = f(x_{n-1})</math> for all <math>n \ge 1</math>. For how ..."
2014-10-23T06:12:20Z
<p>Created page with "==Problem== Let <math>f(x) = 4x - x^{2}</math>. Give <math>x_{0}</math>, consider the sequence defined by <math>x_{n} = f(x_{n-1})</math> for all <math>n \ge 1</math>. For how ..."</p>
<p><b>New page</b></p><div>==Problem==<br />
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Let <math>f(x) = 4x - x^{2}</math>. Give <math>x_{0}</math>, consider the sequence defined by <math>x_{n} = f(x_{n-1})</math> for all <math>n \ge 1</math>. <br />
For how many real numbers <math>x_{0}</math> will the sequence <math>x_{0}, x_{1}, x_{2}, \ldots</math> take on only a finite number of different values?<br />
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<math>\textbf{(A)}\ \text{0}\qquad<br />
\textbf{(B)}\ \text{1 or 2}\qquad<br />
\textbf{(C)}\ \text{3, 4, 5 or 6}\qquad\\<br />
\textbf{(D)}\ \text{more than 6 but finitely many}\qquad\\<br />
\textbf{(E) }\infty</math> <br />
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==Solution==<br />
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== See also ==<br />
{{AHSME box|year=1988|num-b=28|after=Last Question}} <br />
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[[Category: Intermediate Algebra Problems]]<br />
{{MAA Notice}}</div>
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