document.write( "Question 430444: FInd the value of a and b if
\n" ); document.write( "(5^(n+1)-5^(n-1))/(3x5^(3n))=ax5^(bn) \n" ); document.write( "
Algebra.Com's Answer #298981 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Another approach:\r\n" );
document.write( "There are three variables and one equation.\r\n" );
document.write( "Therefore if no other restrictions apply, and\r\n" );
document.write( "if irrational solutions are allowed, then there\r\n" );
document.write( "are infinitely many solutions for instance.  \r\n" );
document.write( "n=-2.5, a=.2139968976, b=-2.5\r\n" );
document.write( "n = -3, a=1, b=-2.097343225\r\n" );
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document.write( "Therefore it is possible only to do one of these\r\n" );
document.write( "three things:\r\n" );
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document.write( "1. Solve for a in terms on n and b\r\n" );
document.write( "2. Solve for b in terms of n and a\r\n" );
document.write( "3. Solve for n in terms of a and b\r\n" );
document.write( "4. Assume the equation is an identity and holds for all values of n\r\n" );
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document.write( "To avoid infinitely many solutions, I will do only the last.  \r\n" );
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document.write( "Since it must hold for all n, if must hold for n=0, \r\n" );
document.write( "so we substitute 0 for n:\r\n" );
document.write( "  \r\n" );
document.write( "\"%285%5E%28n%2B1%29-5%5E%28n-1%29%29%2F%283%2A5%5E%283n%29%29=a%2A5%5E%28bn%29\"\r\n" );
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document.write( "\"%285%5E%280%2B1%29-5%5E%280-1%29%29%2F%283%2A5%5E%283%2A0%29%29=a%2A5%5E%28b%2A0%29\"\r\n" );
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document.write( "\"%285%5E1-5%5E%28-1%29%29%2F3%2A5%5E0=a%2A5%5E0\"\r\n" );
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document.write( "\"%285%5E1-1%2F5%29%29%2F3%2A1=a%2A1\"\r\n" );
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document.write( "\"%285-1%2F5%29%29%2F3=a\"\r\n" );
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document.write( "\"%2825%2F5-1%2F5%29%29%2F3=a\"\r\n" );
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document.write( "\"%2824%2F5%29%2F3=a\"\r\n" );
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document.write( "\"expr%2824%2F5%29%2Aexpr%281%2F3%29=a\"\r\n" );
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document.write( "\"8%2F5=a\"\r\n" );
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document.write( "---\r\n" );
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document.write( "Since it must hold for n=1, we substitute 1 for n, and \"8%2F5\" for a:\r\n" );
document.write( "  \r\n" );
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document.write( "\"%285%5E2-5%5E0%29%2F%283%2A5%5E3%29=expr%288%2F5%29%2A5%5Eb\"\r\n" );
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document.write( "\"%2825-1%29%2F%283%2A5%5E3%29=8%2A%281%2F5%29%2A5%5Eb\"\r\n" );
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document.write( "\"24%2F%283%2A5%5E3%29=8%2A5%5E%28b-1%29\"\r\n" );
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document.write( "\"8%2F%285%5E3%29=8%2A5%5E%28b-1%29\"\r\n" );
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document.write( "\"1%2F5%5E3+=+5%5E%28b-1%29\"\r\n" );
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document.write( "\"5%5E%28-3%29+=+5%5E%28b-1%29\"\r\n" );
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document.write( "Equating exponents of 5\r\n" );
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document.write( "\"-3+=+b-1\"\r\n" );
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document.write( "\"-2+=+b\" \r\n" );
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document.write( "Edwin
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