114 lines
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114 lines
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<h2>Fibonacci golden nuggets</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 137</h3><span style="width:300px;color:#666;">Published on Friday, 12th January 2007, 06:00 pm; Solved by 2613</span></div>
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<div class="problem_content" role="problem">
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<p>Consider the infinite polynomial series A<sub>F</sub>(<i>x</i>) = <i>x</i>F<sub>1</sub> + <i>x</i><sup>2</sup>F<sub>2</sub> + <i>x</i><sup>3</sup>F<sub>3</sub> + ..., where F<sub><i>k</i></sub> is the <i>k</i>th term in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... ; that is, F<sub><i>k</i></sub> = F<sub><i>k</i><img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />1</sub> + F<sub><i>k</i><img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />2</sub>, F<sub>1</sub> = 1 and F<sub>2</sub> = 1.</p>
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<p>For this problem we shall be interested in values of <i>x</i> for which A<sub>F</sub>(<i>x</i>) is a positive integer.</p>
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<table cellpadding='0' cellspacing='0' border='0'>
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<tr>
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<td>Surprisingly A<sub>F</sub>(1/2)</td>
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<td> = </td>
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<td>(1/2).1 + (1/2)<sup>2</sup>.1 + (1/2)<sup>3</sup>.2 + (1/2)<sup>4</sup>.3 + (1/2)<sup>5</sup>.5 + ...</td>
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</tr>
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<tr>
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<td> </td>
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<td> = </td>
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<td>1/2 + 1/4 + 2/8 + 3/16 + 5/32 + ...</td>
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</tr>
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<tr>
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<td> </td>
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<td> = </td>
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<td>2</td>
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</tr>
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</table>
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<p>The corresponding values of <i>x</i> for the first five natural numbers are shown below.</p>
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<div style='text-align:center;'>
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<table cellspacing='0' cellpadding='2' border='1' align='center'>
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<tr style='background-color:#c1daf9;'>
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<td><b><i>x</i></b></td><td width='50'><b>A<sub>F</sub>(<i>x</i>)</b></td>
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</tr>
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<tr>
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<td><img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2<img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />1</td><td>1</td>
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</tr>
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<tr>
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<td>1/2</td><td>2</td>
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</tr>
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<tr>
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<td>(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />13<img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />2)/3</td><td>3</td>
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</tr>
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<tr>
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<td>(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />89<img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />5)/8</td><td>4</td>
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</tr>
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<tr>
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<td>(<img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />34<img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />3)/5</td><td>5</td>
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</tr>
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</table>
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</div>
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<p>We shall call A<sub>F</sub>(<i>x</i>) a golden nugget if <i>x</i> is rational, because they become increasingly rarer; for example, the 10th golden nugget is 74049690.</p>
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<p>Find the 15th golden nugget.</p>
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