document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199949>Question 1199949</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In this continued fraction, the numbers directly in front of the<BR>\n" );
document.write( "addition signs alternate 1, 2, 1, 2, ... infinitely after the first repeated 1. Find the value of the fraction. \r<BR>\n" );
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document.write( "Repeated Fraction Diagram: https://ibb.co/LRJcnLB\r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833934 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=833934\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is the same as writing <font color=red>sqrt(3)</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Let x be equal to the continued fraction given to us.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{x} = 1+\frac{1}{1 + \frac{1}{2+\frac{1}{1+\frac{1}{2+\ldots}}}}\"><BR>\n" );
document.write( "This value is positive because each piece we're adding is positive. \r<BR>\n" );
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document.write( "Now subtract 1 from both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{x}-1=\frac{1}{1 + \frac{1}{2+\frac{1}{1+\frac{1}{2+\ldots}}}}\"><BR>\n" );
document.write( "This is done so that the repeating portions (1,2,1,2,...) are isolated on the right hand side.<BR>\n" );
document.write( "Focus on the left-most part of each row. Ignore the numerators of 1.\r<BR>\n" );
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document.write( "The stuff after \"2+...\", highlighted in the red box, can be replaced with x-1 because of the repeated nature of this sequence.<BR>\n" );
document.write( "The x-1 represents the massive nested fraction block.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+1+=+1%2F%281+%2B+1%2F%282%2Bhighlight%281%2F%281+%2B+1%2F%28matrix%281%2C3%2C%222%22%2C%22%2B%22%2C%22...%22%29%29%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+1+=+1%2F%281+%2B+1%2F%282%2Bhighlight%28%28x-1%29%29%29%29 ALIGN=MIDDLE  ALT=\"x+-+1+=+1%2F%281+%2B+1%2F%282%2Bhighlight%28%28x-1%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+1+=+1%2F%281+%2B+1%2F%282%2B%28x-1%29%29%29 ALIGN=MIDDLE  ALT=\"x+-+1+=+1%2F%281+%2B+1%2F%282%2B%28x-1%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+1+=+1%2F%281+%2B+1%2F%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"x+-+1+=+1%2F%281+%2B+1%2F%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+1+=+1%2F%28%28x%2B1%29%2F%28x%2B1%29+%2B+1%2F%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"x+-+1+=+1%2F%28%28x%2B1%29%2F%28x%2B1%29+%2B+1%2F%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+1+=+1%2F%28%28x%2B1%2B1%29%2F%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"x+-+1+=+1%2F%28%28x%2B1%2B1%29%2F%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+1+=+1%2F%28%28x%2B2%29%2F%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"x+-+1+=+1%2F%28%28x%2B2%29%2F%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+1+=+%28x%2B1%29%2F%28x%2B2%29 ALIGN=MIDDLE  ALT=\"x+-+1+=+%28x%2B1%29%2F%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28x%2B2%29+=+x%2B1 ALIGN=MIDDLE  ALT=\"%28x-1%29%28x%2B2%29+=+x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-2+=+x%2B1 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-2+=+x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-x+=+1%2B2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-x+=+1%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+1%2B2 ALIGN=MIDDLE  ALT=\"x%5E2+=+1%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+3 ALIGN=MIDDLE  ALT=\"x%5E2+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%22%22%2B-+sqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+%22%22%2B-+sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Keep in mind that x cannot be negative since each of the infinitely many fractional pieces we're adding are positive.\r<BR>\n" );
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document.write( "Positive + positive = positive\r<BR>\n" );
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document.write( "This rules out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-sqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we're left with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sqrt%283%29 ALIGN=MIDDLE  ALT=\"x+=+sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the final answer.\r<BR>\n" );
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document.write( "Therefore, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 1+\frac{1}{1 + \frac{1}{2+\frac{1}{1+\frac{1}{2+\ldots}}}} = \sqrt{3}\">\r<BR>\n" );
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document.write( "-------------------------------------\r<BR>\n" );
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document.write( "You can use coding software to generate each <a  rel=nofollow HREF= \"https://mathworld.wolfram.com/Convergent.html\">convergent</a> which will help partially verify the answer. <BR>\n" );
document.write( "I have done so as indicated below:<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "1 + 1/1 = 2\r\n" );
document.write( "1 + 1/(1+1/2) = 1.66666666666667\r\n" );
document.write( "1 + 1/(1+1/(2 + 1/(1 + 1/2))) = 1.72727272727273\r\n" );
document.write( "1 + 1/(1+1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/2))))) = 1.73170731707317\r\n" );
document.write( "1 + 1/(1+1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/2))))))) = 1.73202614379085\r\n" );
document.write( "sqrt(3) = 1.73205080756888\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "Admittedly the precision isn't too great, but we have 4 decimal digits matching. <BR>\n" );
document.write( "I.e. we're able to get to 1.7320, then the rest of the decimal digits are incorrect.<BR>\n" );
document.write( "This process is continued until reaching the desired level of precision.<BR>\n" );
document.write( "Each decimal value is approximate.\r<BR>\n" );
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document.write( "Another way to numerically approximate x is to compute the pieces in reverse order.<BR>\n" );
document.write( "Rather than go left to right, I'm working right to left. Furthermore, I'm starting on the inside and working toward the outside of this massive nested fraction. This is a recursive process.<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Let ans = 2\r\n" );
document.write( "1 + 1/ans = 1.5\r\n" );
document.write( "2 + 1/ans = 2.66666666666667\r\n" );
document.write( "1 + 1/ans = 1.375\r\n" );
document.write( "2 + 1/ans = 2.72727272727273\r\n" );
document.write( "1 + 1/ans = 1.36666666666667\r\n" );
document.write( "2 + 1/ans = 2.73170731707317\r\n" );
document.write( "1 + 1/ans = 1.36607142857143\r\n" );
document.write( "2 + 1/ans = 2.73202614379085\r\n" );
document.write( "1 + 1/ans = 1.36602870813397\r\n" );
document.write( "2 + 1/ans = 2.73204903677758\r\n" );
document.write( "1 + 1/ans = 1.36602564102564\r\n" );
document.write( "2 + 1/ans = 2.73205068043172\r\n" );
document.write( "1 + 1/ans = 1.36602542081759\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "I used software to generate the list above.<BR>\n" );
document.write( "For any given line, each 'ans' refers to the answer of the previous line.\r<BR>\n" );
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document.write( "The stopping criteria is anything starting with \"1+1/ans\" to match what was the denominator for the upper-most level of x-1.<BR>\n" );
document.write( "If we stopped at 1.36602542081759, then the reciprocal of this is 1/1.36602542081759 = 0.73205079844083<BR>\n" );
document.write( "Add on 1 to get 1.73205079844083 which is somewhat close to sqrt(3) = 1.73205080756888\r<BR>\n" );
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document.write( "Compare the digits<BR>\n" );
document.write( "<font color=red>1.732050</font>79844083 ...... computed from continued fraction<BR>\n" );
document.write( "<font color=red>1.732050</font>80756888 ...... calculator's display of sqrt(3)<BR>\n" );
document.write( "The stuff in red is what matches.\r<BR>\n" );
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document.write( "-------------------------------------\r<BR>\n" );
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document.write( "Further Reading:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://sites.millersville.edu/bikenaga/number-theory/periodic-continued-fractions/periodic-continued-fractions.html\">https://sites.millersville.edu/bikenaga/number-theory/periodic-continued-fractions/periodic-continued-fractions.html</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://pi.math.cornell.edu/~gautam/ContinuedFractions.pdf\">https://pi.math.cornell.edu/~gautam/ContinuedFractions.pdf</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://mathworld.wolfram.com/ContinuedFraction.html\">https://mathworld.wolfram.com/ContinuedFraction.html</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/Continued_fraction\">https://en.wikipedia.org/wiki/Continued_fraction</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.planetmath.org/TableOfContinuedFractionsOfSqrtnFor1N102\">https://www.planetmath.org/TableOfContinuedFractionsOfSqrtnFor1N102</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://brilliant.org/wiki/continued-fractions/\">https://brilliant.org/wiki/continued-fractions/</a><BR>\n" );
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