document.write( "Question 1066671: Find the value of
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Algebra.Com's Answer #681896 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "If I understand (guess) your problem correctly, you are asking about this infinite continued fraction\r\n" );
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document.write( "1\r\n" );
document.write( "--------------------------\r\n" );
document.write( "      1\r\n" );
document.write( "1 + ----------------------\r\n" );
document.write( "           1\r\n" );
document.write( "      2 + ----------------\r\n" );
document.write( "                 1 \r\n" );
document.write( "           1 + -----------\r\n" );
document.write( "                      1\r\n" );
document.write( "                2 + ------\r\n" );
document.write( "                        \r\n" );
document.write( "                    1 + --\r\n" );
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document.write( "The standard way of solving such problems is THIS:\r\n" );
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document.write( "Let \"x\" be the value (now unknown) of this infinite continued fraction.\r\n" );
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document.write( "Then we can write the fraction in this form\r\n" );
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document.write( "      1 \r\n" );
document.write( "x = ------------------------\r\n" );
document.write( "            1  \r\n" );
document.write( "      1 + ------------------\r\n" );
document.write( "           2 + x\r\n" );
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document.write( "and it is just  an equation in the FINITE fraction form.\r\n" );
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document.write( "Simplify it using standard rules, and you will get\r\n" );
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document.write( "      2 + x                2 + x\r\n" );
document.write( "x = ---------------  =  --------------\r\n" );
document.write( "      (2 + x) + 1          3 + x\r\n" );
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document.write( "Multiply both sides by 3+x to rid of the denominator. You will get\r\n" );
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document.write( "x*(3+x) = 2 + x,\r\n" );
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document.write( "x^2 + 3x = 2 + x,\r\n" );
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document.write( "x^2 + 2x - 2 = 0,\r\n" );
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document.write( " =  =  = .\r\n" );
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document.write( "Only positive solution works.\r\n" );
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document.write( "Therefore, x =  = 0.732051 . . . .\r\n" );
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document.write( "Approximately, x = 0.732051.\r\n" );
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