document.write( "Question 1210195: If a,b,c are in Harmonic Progression,prove that 1/a + 1/(b+c), 1/b + 1(c+a), 1/c + 1(a+b) are also in Harmonic Progression \n" ); document.write( "
Algebra.Com's Answer #851531 by mccravyedwin(407)\"\" \"About 
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document.write( "If a,b,c are in Harmonic Progression,\r\n" );
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document.write( "That means that 1/a, 1/b, 1/c are in arithmetic progression, i.e.\r\n" );
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document.write( "\"1%2Fb-1%2Fa=1%2Fc-1%2Fb\"\r\n" );
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document.write( "\"ac-bc=ab-ac\"\r\n" );
document.write( "\"2ac=ab%2Bbc\"\r\n" );
document.write( "\"2ac=b%28a%2Bc%29\"\r\n" );
document.write( "\"b=%282ac%29%2F%28a%2Bc%29\"\r\n" );
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document.write( "We are to show that\r\n" );
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document.write( "(1)   \"1%5E%22%22%2F%281%2Fb+%2B+1%28c%2Ba%29%29-+1%5E%22%22%2F%281%2Fa+%2B+1%2F%28b%2Bc%29%29\"\"%22%22=%22%22\"\"1%5E%22%22%2F%281%2Fc+%2B+1%28a%2Bb%29+%29-1%5E%22%22%2F%281%2Fb+%2B+1%28c%2Ba%29%29\"\r\n" );
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document.write( "All terms are of the form \"1%5E%22%22%2F%281%2Fx+%2B+1%2F%28y%2Bz%29%29\"\r\n" );
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document.write( "Multiply top and bottom by x(y+z)\r\n" );
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document.write( "\"%28x%28y%2Bz%29%29%2F%28y%2Bz%2Bx%29%29\"\r\n" );
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document.write( "So (1) becomes\r\n" );
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document.write( "(2)   \"%28b%28c%2Ba%29%29%2F%28c%2Ba%2Bb%29-%28a%28b%2Bc%29%29%2F%28b%2Bc%2Ba%29\"\"%22%22=%22%22\"\"%28c%28a%2Bb%29%29%2F%28a%2Bb%2Bc%29-%28b%28c%2Ba%29%29%2F%28c%2Ba%2Bb%29\" \r\n" );
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document.write( "(2) will be true if and only if\r\n" );
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document.write( "\"%282b%28c%2Ba%29%29%2F%28c%2Ba%2Bb%29-%28a%28b%2Bc%29%29%2F%28b%2Bc%2Ba%29\"\"%22%22=%22%22\"\"%28c%28a%2Bb%29%29%2F%28a%2Bb%2Bc%29\" \r\n" );
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document.write( "which will be true if and only if\r\n" );
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document.write( "\"2b%28c%2Ba%29-a%28b%2Bc%29\"\"%22%22=%22%22\"\"c%28a%2Bb%29\"\r\n" );
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document.write( "which will be true if and only if \r\n" );
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document.write( "\"2bc%2B2ab-ab-ac\"\"%22%22=%22%22\"\"ac%2Bbc\"\r\n" );
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document.write( "which will be true if and only if\r\n" );
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document.write( "\"bc%2Bab-2ac\"\"%22%22=%22%22\"\"0\"\r\n" );
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document.write( "Substitute \"b=%282ac%29%2F%28a%2Bc%29\"\r\n" );
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document.write( "\"%28%282ac%29%2F%28a%2Bc%29%29c%2Ba%28%282ac%29%2F%28a%2Bc%29%29-2ac\"\"%22%22=%22%22\"\"0\"\r\n" );
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document.write( "which will be true if and only if\r\n" );
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document.write( "\"2ac%5E2%2B2a%5E2c-2ac%28a%2Bc%29+=+0\"\r\n" );
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document.write( "Which will be true if and only if \r\n" );
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document.write( "\"c%2Ba+-+%28a+%2B+c%29+=+0\"\r\n" );
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document.write( "And since this is true, the problem is proved.\r\n" );
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document.write( "Edwin
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