174 lines
7.7 KiB
HTML
174 lines
7.7 KiB
HTML
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<title>Problem 330 - Project Euler</title>
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<h2>Euler's Number</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 330</h3><span style="width:300px;color:#666;">Published on Sunday, 27th March 2011, 05:00 am; Solved by 314</span></div>
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<div class="problem_content" role="problem">
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An infinite sequence of real numbers <var>a</var>(<var>n</var>) is defined for all integers <var>n</var> as follows:
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<div align='center'><img src="project/images/p_330_formula.gif" /></div>
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<p>For example,<br />
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<table class="formula"><tr>
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<td><var>a</var>(0) = </td>
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<td><table class="frac">
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<tr><td>1</td></tr>
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<tr><td class="overline">1!</td></tr>
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</table>
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</td>
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<td>+</td>
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<td><table class="frac">
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<tr><td>1</td></tr>
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<tr><td class="overline">2!</td></tr>
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</table>
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</td>
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<td>+</td>
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<td><table class="frac">
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<tr><td>1</td></tr>
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<tr><td class="overline">3!</td></tr>
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</table>
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</td>
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<td>+ ... = e <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 1 </td>
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</tr></table>
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<table class="formula"><tr>
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<td><var>a</var>(1) = </td>
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<td><table class="frac">
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<tr><td>e <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 1</td></tr>
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<tr><td class="overline">1!</td></tr>
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</table>
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</td>
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<td>+</td>
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<td><table class="frac">
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<tr><td>1</td></tr>
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<tr><td class="overline">2!</td></tr>
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</table>
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</td>
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<td>+</td>
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<td><table class="frac">
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<tr><td>1</td></tr>
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<tr><td class="overline">3!</td></tr>
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</table>
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</td>
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<td>+ ... = 2e <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 3 </td>
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</tr></table>
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<table class="formula"><tr>
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<td><var>a</var>(2) = </td>
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<td><table class="frac">
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<tr><td>2e <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 3</td></tr>
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<tr><td class="overline">1!</td></tr>
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</table>
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</td>
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<td>+</td>
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<td><table class="frac">
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<tr><td>e <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 1</td></tr>
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<tr><td class="overline">2!</td></tr>
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</table>
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</td>
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<td>+</td>
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<td><table class="frac">
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<tr><td>1</td></tr>
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<tr><td class="overline">3!</td></tr>
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</table>
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</td>
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<td>+ ... =</td>
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<td><table class="frac">
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<tr><td>7</td></tr>
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<tr><td class="overline">2</td></tr>
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</table>
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</td>
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<td>e <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 6 </td>
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</tr></table>
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with e = 2.7182818... being Euler's constant.
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</p>
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<p>
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<table class="formula"><tr>
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<td>It can be shown that <var>a</var>(<var>n</var>) is of the form
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</td>
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<td><table class="frac">
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<tr><td>A(<var>n</var>) e + B(<var>n</var>)</td></tr>
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<tr><td class="overline"><var>n</var>!</td></tr>
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</table>
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</td>
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<td>for integers A(<var>n</var>) and B(<var>n</var>).
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</td>
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</tr></table>
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<table class="formula"><tr>
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<td>For example <var>a</var>(10) =
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</td>
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<td><table class="frac">
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<tr><td>328161643 e <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 652694486</td></tr>
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<tr><td class="overline">10!</td></tr>
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</table>
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</td>
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<td>.</td>
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</tr></table>
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</p>
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<P>
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Find A(10<sup>9</sup>) + B(10<sup>9</sup>) and give your answer mod 77 777 777.
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</P>
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