document.write( "Question 461625: hi am stuck on a question on theory of numbers\r
\n" ); document.write( "\n" ); document.write( "a, b, c, c, are four unequal positive numbers. by using the theorem (if a=b, then a+c=b+c and ac=bc) and considering the first pair of numbers 1/2(a+b) and 1/2(c+d) and then separately, the pairs a,b and c,d prove that\r
\n" ); document.write( "\n" ); document.write( "1/4(a+b+c+d)>(abcd)^1/4\r
\n" ); document.write( "\n" ); document.write( "Deduce by considering the four unequal numbers a,b,c and 1/3(a+b+c)that\r
\n" ); document.write( "\n" ); document.write( "1/3(a+b+c)>(abc)^1/3\r
\n" ); document.write( "\n" ); document.write( "what happens to the last inequality (i) if a=b=c
\n" ); document.write( " (ii)if a=b≠c?
\n" ); document.write( "suggest a generalization of the results proved above?\r
\n" ); document.write( "\n" ); document.write( "Phew this is one of the toughest questions i have ever come across? \n" ); document.write( "
Algebra.Com's Answer #316757 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Since the inequality we are to prove for the letters\r\n" );
document.write( "representing unequal positive numbers:\r\n" );
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document.write( "and one of the other parts is\r\n" );
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document.write( "Then we might suppose that the first thing we need to\r\n" );
document.write( "prove is that for any different positive numbers a and b \r\n" );
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document.write( "The technique I will use is to assume that it is false,\r\n" );
document.write( "and then reach a contradiction.  \r\n" );
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document.write( "So for contradiction we will assume that for \"0%3Ea%3C%3Eb%3E0\",  \r\n" );
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document.write( "Multiply both sides by 2\r\n" );
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document.write( "Square both sides:\r\n" );
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document.write( "Subtract 4ab from both sides:\r\n" );
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document.write( "Factor the left side:\r\n" );
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document.write( "\"drawing%28175%2C60%2C0%2C10%2C-10%2C7%2C+locate%281%2C3%2C%0D%0A%28a-b%29%5E2%3C=0+%29+%29\"\r\n" );
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document.write( "This is false since the left side is positive,\r\n" );
document.write( "since a and b are different.\r\n" );
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document.write( "So we have reached a contradiction and therefore\r\n" );
document.write( "we have proved that \r\n" );
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document.write( "Note that the two sides are equal if a=b.\r\n" );
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document.write( "We have also proved by letting a=c and b=d.\r\n" );
document.write( "that for \"0%3Ec%3C%3Ed%3E0\" \r\n" );
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document.write( "Down below we will need this for positive x,y: \r\n" );
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document.write( "with equality holding if and only if x=y.\r\n" );
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document.write( "We do this to avoid getting letters confused.\r\n" );
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document.write( "Next we will prove that for a,b,c,d > 0, all different,\r\n" );
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document.write( "We start with\r\n" );
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document.write( "Multiply both sides by 2, and we have:\r\n" );
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document.write( "Add the two inequalities:\r\n" );
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document.write( "Factor out 2 on the right\r\n" );
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document.write( "Make the left side into the left side of what\r\n" );
document.write( "we have to prove by multiplying thru hy 1/4\r\n" );
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document.write( "Now we will show that the right side is greater than or equal\r\n" );
document.write( "to \"drawing%28100%2C60%2C0%2C10%2C-10%2C7%2C+locate%281%2C3%2C+%28abcd%29%5E%281%2F4%29%29+%29\"\r\n" );
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document.write( "We recall from above that for positive x, y,\r\n" );
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document.write( "with equality holding if and only if x=y\r\n" );
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document.write( "We let x = \"drawing%28100%2C60%2C0%2C10%2C-10%2C7%2C+locate%281%2C3%2C+%28ab%29%5E%281%2F2%29%29+%29\"\r\n" );
document.write( "and y = \"drawing%28100%2C60%2C0%2C10%2C-10%2C7%2C+locate%281%2C3%2C+%28cd%29%5E%281%2F2%29%29+%29\"\r\n" );
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document.write( "[Notice that we had to include the possibility of equality \r\n" );
document.write( "since even though a,b,c,d are all distinct, that does not\r\n" );
document.write( "guarantee that \"drawing%28100%2C60%2C0%2C10%2C-10%2C7%2C+locate%281%2C3%2C+%28ab%29%5E%281%2F2%29%29+%29\"\r\n" );
document.write( " and \"drawing%28100%2C60%2C0%2C10%2C-10%2C7%2C+locate%281%2C3%2C+%28cd%29%5E%281%2F2%29%29+%29\"\r\n" );
document.write( " are distinct, e.g. if a=1,b=6,c=2,d=3, they are not distinct.]\r\n" );
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document.write( "So we have proved:\r\n" );
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document.write( "which is what we had to proved.\r\n" );
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document.write( "For different positive a,b,c, we need to prove\r\n" );
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document.write( "Sorry, I haven't figured out how to do this one yet.\r\n" );
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document.write( "The generalization.  If all ai are \r\n" );
document.write( "positive numbers, then\r\n" );
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document.write( "and equality holds only when all the ai are equal.\r\n" );
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document.write( "Edwin
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