document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=982143>Question 982143</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 #603005 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52798)</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=603005\">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 positive, unequal real numbers. \r<BR>\n" );
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document.write( "Then 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; and 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;\" >.\r<BR>\n" );
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document.write( "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=%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( "Let us start with this 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,\r<BR>\n" );
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document.write( "which is always true for any real unequal numbers &nbsp;<B>a</B>&nbsp; and &nbsp;<B>b</B>, &nbsp;because the left part is the square of the real number &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;which is unequal to zero.\r<BR>\n" );
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document.write( "Rewrite this inequality in this way step by step:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+-+2sqrt%28a%29%2Asqrt%28b%29+%2B+b ALIGN=MIDDLE  ALT=\"a+-+2sqrt%28a%29%2Asqrt%28b%29+%2B+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;&nbsp;&nbsp;&nbsp;&nbsp;(apply the formula of the square of a difference)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b ALIGN=MIDDLE  ALT=\"a+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%28a%29%2Asqrt%28b%29 ALIGN=MIDDLE  ALT=\"2sqrt%28a%29%2Asqrt%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(move the term &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-+2sqrt%28a%29%2Asqrt%28b%29 ALIGN=MIDDLE  ALT=\"-+2sqrt%28a%29%2Asqrt%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; from the left side to the right with the opposite sign)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%2F2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%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( "The last inequality is exactly what you have to prove. \r<BR>\n" );
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