document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185523>Question 1185523</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Insert the indicated number of harmonic means given the first and last terms. Show the solution.\r<BR>\n" );
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document.write( "1/2 and 1/6   [3] or insert 3 \r<BR>\n" );
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document.write( "2/3 and 2/9   [2] or insert 2\r<BR>\n" );
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document.write( "5/2 and 5/27  [4] or insert 4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #816654 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=816654\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "I will do the first progression only-- insert 3 harmonic means between 1/2 and 1/6.\r<BR>\n" );
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document.write( "If five numbers are in harmonic progression then their reciprocals are in arithmetic progression.  \r<BR>\n" );
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document.write( "Let those numbers in AP be 2, a, b, c, 6.\r<BR>\n" );
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document.write( "Then it's quite clear that the numbers in between are 2, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%283%29 ALIGN=MIDDLE  ALT=\"red%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+red%284%29 ALIGN=MIDDLE  ALT=\"+red%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%285%29 ALIGN=MIDDLE  ALT=\"red%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 6.\r<BR>\n" );
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document.write( "Therefore the three harmonic means between 1/2 and 1/6 are 1/3, 1/4, and 1/5.\r<BR>\n" );
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