document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=876826>Question 876826</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mid-points of the side of an equilateral<BR>\n" );
document.write( "triangle of side 18 cm are joined to form<BR>\n" );
document.write( "another triangle, whose mid-points are<BR>\n" );
document.write( "further joined to form a different triangle and<BR>\n" );
document.write( "this process is repeated indefinitely. The sum<BR>\n" );
document.write( "of the perimeters of all the triangles will be </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #529023 by </B><A HREF=/tutors/aboutme.mpl?userid=fcabanski><B>fcabanski(1391)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=fcabanski><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=529023\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The perimeter of the original triangle is 3*18=54cm.  The triangle formed by the first midpoint connection has sides of 9cm.  Its perimeter is 3*9cm = 27cm.  The third triangle has sides of 4.5cm.  Its perimeter is 3*4.5cm = 13.5cm.<P><BR>\n" );
document.write( "The total perimeter is 3*(18+9+4.5+...)<P>\r<BR>\n" );
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document.write( "18+9+4.5... is a geometric series 18/(2^n) where n is 0 to infinity.<P><BR>\n" );
document.write( "The sum of an infinite geometric series xr^0+xr^1+xr^2+...xr^n+... converges to a certain value which is x/(1-r).  In this case x=18 and r=1/2.  So the sum is 18/(1/2) = 18*2 =36.<P><BR>\n" );
document.write( "The total perimeter is 3*36 = 108 cm. </SPAN>\n" );
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