78 lines
5.1 KiB
HTML
78 lines
5.1 KiB
HTML
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<h2>Square root convergents</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 57</h3><span style="width:300px;color:#666;">Published on Friday, 21st November 2003, 06:00 pm; Solved by 18642</span></div>
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<p>It is possible to show that the square root of two can be expressed as an infinite continued fraction.</p>
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<p style='text-align:center;'><img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' /> 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...</p>
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<p>By expanding this for the first four iterations, we get:</p>
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<p>1 + 1/2 = 3/2 = 1.5<br />
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1 + 1/(2 + 1/2) = 7/5 = 1.4<br />
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1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...<br />
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1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...<br /></p>
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<p>The next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.</p>
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<p>In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?</p>
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