document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=984031>Question 984031</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that the geometric mean of two positive, unequal numbers is less than their arithmetic mean. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #604838 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52792)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=604838\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Let &nbsp;<B>a</B>&nbsp; and &nbsp;<B>b</B>&nbsp; be two real positive unequal numbers.\r<BR>\n" );
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document.write( "Their arithmetic mean is &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a+%2B+b%29%2F2 ALIGN=MIDDLE  ALT=\"%28a+%2B+b%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;Their geometric mean is &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%2Ab%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%2Ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. &nbsp;We need to prove that \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%2Ab%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%2Ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > < <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a+%2B+b%29%2F2 ALIGN=MIDDLE  ALT=\"%28a+%2B+b%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let us start with inequality\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28a%29+-+sqrt%28b%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28a%29+-+sqrt%28b%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > > 0. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)\r<BR>\n" );
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document.write( "This inequality is true because the difference &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%29+-+sqrt%28b%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%29+-+sqrt%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; is not zero and the square of a non-zero real number is positive. &nbsp;Now, &nbsp;expand &nbsp;(1)&nbsp; as \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28a%29+-+sqrt%28b%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28a%29+-+sqrt%28b%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28a%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28a%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%28a%29%2Asqrt%28b%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%28a%29%2Asqrt%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28b%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28b%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%28a%29%2Asqrt%28b%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%28a%29%2Asqrt%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Thus the original inequality &nbsp;(1)&nbsp; is equivalent to  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%28a%29%2Asqrt%28b%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%28a%29%2Asqrt%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;&nbsp;&nbsp;or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >  + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%28a%29%2Asqrt%28b%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%28a%29%2Asqrt%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;&nbsp;&nbsp;or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a+%2B+b%29%2F2 ALIGN=MIDDLE  ALT=\"%28a+%2B+b%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%2Ab%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%2Ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "It is exactly what has to be proved. \r<BR>\n" );
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document.write( " </SPAN>\n" );
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