document.write( "Question 1091490: if a,b,c are in AP, aalnd a,mb,c are in GP then a, m^2b, c are in HP \n" ); document.write( "
Algebra.Com's Answer #706092 by Edwin McCravy(20056)\"\" \"About 
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document.write( "If R,S,T are in AP, then S-R = T-S, or \r\n" );
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document.write( "(1)  2S = R+T\r\n" );
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document.write( "If U,V,W are in GP, then \"V%2FU\"\"%22%22=%22%22\"\"W%2FV\", or \r\n" );
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document.write( "(2)  UW = V2\r\n" );
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document.write( "If X,Y,Z are in HP, then their reciprocals are in AP, so\r\n" );
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document.write( "\"1%2FY-1%2FX\"\"%22%22=%22%22\"\"1%2FZ-1%2FY\", \r\n" );
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document.write( "Multiply through by LCD = XYZ\r\n" );
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document.write( "XZ-YZ = XY-XZ\r\n" );
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document.write( "(3)  2XZ = XY+YZ\r\n" );
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document.write( "Since a,b,c are in AP, then we substitute R=a, S=b, T=c in (1)\r\n" );
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document.write( "(4)  2b = a+c               <--we know this\r\n" );
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document.write( "Since  a, mb, c are in GP, then we substitute U=a, V=mb, W=c in (2)\r\n" );
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document.write( "(5)  ac = m2b2            <--we know this\r\n" );
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document.write( "We are to prove that a, m2b, c are in HP.  So to find \r\n" );
document.write( "out what we need to prove, we substitute X=a, Y=m2b, Z=c in (3)\r\n" );
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document.write( "(6)  2ac = am2b + m2bc    <--we are to prove this\r\n" );
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document.write( "So we will start with 2ac, and show that it is equal to am2b + m2bc\r\n" );
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document.write( "By (5),    \r\n" );
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document.write( "2ac = 2m2b2 \r\n" );
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document.write( "2ac = m2∙b∙2b, use (4) to substitute a+c for 2b \r\n" );
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document.write( "2ac = m2∙b∙(a+c), distribute:\r\n" );
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document.write( "1ac = m2∙b∙a + m2∙b∙c\r\n" );
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document.write( "That's the same as\r\n" );
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document.write( "2ac = am2b + m2bc\r\n" );
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document.write( "which is (6), which is what we were to prove.\r\n" );
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document.write( "Edwin
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