document.write( "Question 916516: Find the value of n so that\r
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Algebra.Com's Answer #556120 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
The geometric mean between positive a and b is \"sqrt%28ab%29\"\r\n" );
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document.write( "\"+%28+a%5E%28n%2B1%29+%2B+b%5E%28n%2B1%29+%29%2F%28+a%5En+%2B+b%5En+%29\"\"%22%22=%22%22\"\"sqrt%28ab%29\"\r\n" );
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document.write( "Multiply both sides by \"%28+a%5En+%2B+b%5En+%29\",\r\n" );
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document.write( "\"++a%5E%28n%2B1%29+%2B+b%5E%28n%2B1%29+\"\"%22%22=%22%22\"\"sqrt%28ab%29%28+a%5En+%2B+b%5En+%29\" \r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Ca%5E%28n%2B1%29+%2B+b%5E%28n%2B1%29%29+\"\"%22%22=%22%22\"\"matrix%282%2C1%2C%22%22%2C%28ab%29%5E%281%2F2%29%28+a%5En+%2B+b%5En+%29%29\"\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Ca%5E%28n%2B1%29+%2B+b%5E%28n%2B1%29%29+\"\"%22%22=%22%22\"\"matrix%282%2C1%2C%22%22%2Ca%5E%281%2F2%29b%5E%281%2F2%29%28+a%5En+%2B+b%5En+%29%29\"\r\n" );
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document.write( "Distributing:\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Ca%5E%28n%2B1%29+%2B+b%5E%28n%2B1%29%29+\"\"%22%22=%22%22\"\r\n" );
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document.write( "Since this must be an identity for all postive a and b\r\n" );
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document.write( "The first term on the left must be identical to the second term on the right\r\n" );
document.write( "and the second term on the left must be identical to the first term on right.\r\n" );
document.write( "Let's equate the first term on the left and the second term on the right:\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Ca%5E%28n%2B1%29%29+\"\"%22%22=%22%22\"\"matrix%282%2C1%2C%22%22%2C+a%5E%281%2F2%29b%5E%28n%2B1%2F2%29+%29%29\"\r\n" );
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document.write( "That will be an identity only if the exponents of a are equal and the exponent\r\n" );
document.write( "of b is 0 since b to the 0 power is 1, that is\r\n" );
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document.write( "\"n%2B1=1%2F2\" and \r\n" );
document.write( "\"matrix%282%2C1%2C%22%22%2Cb%5E%28n%2B1%2F2%29=1%29\"\r\n" );
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document.write( "And that is indeed true if \"n=-1%2F2\" for then we have\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Ca%5E%28-1%2F2%2B1%29%29+\"\"%22%22=%22%22\"\"matrix%282%2C1%2C%22%22%2C+a%5E%281%2F2%29b%5E%28-1%2F2%2B1%2F2%29+%29%29\" \r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Ca%5E%281%2F2%29%29+\"\"%22%22=%22%22\"\"matrix%282%2C1%2C%22%22%2C+a%5E%281%2F2%29b%5E%280%29+%29%29\"\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Ca%5E%281%2F2%29%29+\"\"%22%22=%22%22\"\"matrix%282%2C1%2C%22%22%2C+a%5E%281%2F2%29%281%29+%29%29\"\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Ca%5E%281%2F2%29%29+\"\"%22%22=%22%22\"\"matrix%282%2C1%2C%22%22%2C+a%5E%281%2F2%29+%29%29\"\r\n" );
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document.write( "It's the same when we equate the second term on the left with the first term on right.  \r\n" );
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document.write( "Thus \"n=-1%2F2\"\r\n" );
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document.write( "Edwin
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