document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=896477>Question 896477</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rope of lenght 2m is divided into three pieces whose lengths are in a geometric sequence. The longest piece is twice as long as the shortest piece. find the common ratio of the sequence and the exact length of the shortest piece of rope ( without a GCD). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #543533 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=543533\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the nth term of a geometric series is given by the formula of:\r<BR>\n" );
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document.write( "An = A1 * r^(n-1)\r<BR>\n" );
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document.write( "the sum of this geometric series must be equal to 2 because that's the overall length of your rope.\r<BR>\n" );
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document.write( "your problem states that the longest piece is 2 times the length of the shortest piece.\r<BR>\n" );
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document.write( "this means that, if the shortest piece is equal to x, then the longest piece must be equal to 2x.\r<BR>\n" );
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document.write( "your rope is being divided into 3 segments, so the value of n in your series will be equal to 3.\r<BR>\n" );
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document.write( "your sequence becomes:\r<BR>\n" );
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document.write( "A1 = x<BR>\n" );
document.write( "A2 = x*r<BR>\n" );
document.write( "A3 = x*r^2 = 2x *****\r<BR>\n" );
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document.write( "***** A3 is equal to 2x and 2x is equal to x*r^2\r<BR>\n" );
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document.write( "your equation to solve from A3 is 2x = x^r^2\r<BR>\n" );
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document.write( "divide both sides of this equation by x to get:\r<BR>\n" );
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document.write( "2x/x = r^2\r<BR>\n" );
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document.write( "solve for r^2 to get r^2 = 2\r<BR>\n" );
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document.write( "solve for r to get r = sqrt(2)\r<BR>\n" );
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document.write( "your common ratio is sqrt(2)\r<BR>\n" );
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document.write( "your geometric series becomes:\r<BR>\n" );
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document.write( "A1 = x<BR>\n" );
document.write( "A2 = x*sqrt(2)<BR>\n" );
document.write( "A3 = x*sqrt(2)^2\r<BR>\n" );
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document.write( "the formula for the sum of a geometric series is:\r<BR>\n" );
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document.write( "Sn = A1 * (1 - r^n) / (1-r)\r<BR>\n" );
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document.write( "you know that Sn = 2 because that's the length of your rope.\r<BR>\n" );
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document.write( "you know that r = sqrt(2) because you just solved for that.\r<BR>\n" );
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document.write( "you know that A1 = x because that's the letter you assigned to represent the value of A1.\r<BR>\n" );
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document.write( "the Sn formula becomes:\r<BR>\n" );
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document.write( "2 = x * (1 - sqrt(2)^3) / (1 - sqrt(2))\r<BR>\n" );
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document.write( "multiply both sides of this formula by (1 - sqrt(2)) to get:\r<BR>\n" );
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document.write( "2 * (1 - sqrt(2)) = x * (1 - sqrt(2)^3)\r<BR>\n" );
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document.write( "divide both sides of this formula by (1 - sqrt(2)^3) to get:\r<BR>\n" );
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document.write( "2 * (1 - sqrt(2)) / (1 - sqrt(2)^3) = x\r<BR>\n" );
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document.write( "that's the value of x which is the value of A1 which is the smallest length of rope.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "r = sqrt(2)\r<BR>\n" );
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document.write( "A1 = 2 * (1 - sqrt(2)) / (1 - sqrt(2)^3)\r<BR>\n" );
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document.write( "i'll use my calculator to confirm for you that the sum of the geometric series for n = 1 to 3 is equal to 2 which is the length of the rope, and that the smallest length of the rope is equal to A1 as shown above.\r<BR>\n" );
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document.write( "using my calculator, and using the formula of An = A1 * r^(n-1) for n = 1 to 3, i get:\r<BR>\n" );
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document.write( "A1 = .4530818393<BR>\n" );
document.write( "A2 = .640754482<BR>\n" );
document.write( "A3 = .9061636786\r<BR>\n" );
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document.write( "sum(A1 to A3) = 2\r<BR>\n" );
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document.write( "use your calculator to solve for A1 and you will see that:\r<BR>\n" );
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document.write( "A1 = 2 * (1 - sqrt(2)) / (1 - sqrt(2)^3) = .4530818393\r<BR>\n" );
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document.write( "your solutions are:\r<BR>\n" );
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document.write( "r = sqrt(2)\r<BR>\n" );
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document.write( "A1 = 2 * (1 - sqrt(2)) / (1 - sqrt(2)^3)\r<BR>\n" );
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document.write( " </SPAN>\n" );
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