document.write( "Question 982823: using mathematicsl induction prove that 3^n+7^n-2 is divisible by 8 \n" ); document.write( "
Algebra.Com's Answer #603609 by Edwin McCravy(20056)\"\" \"About 
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using mathematicsl induction prove that \"3%5En%2B7%5En-2\" is divisible by 8
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document.write( "\"f%28n%29+=+3%5En%2B7%5En-2\"\r\n" );
document.write( "\"f%281%29+=+3%5E1%2B7%5E1-2=3%2B7-2=8\" is divisible by 8.\r\n" );
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document.write( "We only need to show that the difference f(n+1)-f(n) is divisible by 8. \r\n" );
document.write( "For if that difference is divisible by 8 then the assumption that f(n) \r\n" );
document.write( "is divisible by 8 will show that f(n+1) is divisible by 8 because the \r\n" );
document.write( "sum of the difference and f(n) will be f(n+1) and the sum of two \r\n" );
document.write( "multiples of 8 is a multiple of 8. \r\n" );
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document.write( "\"f%28n%2B1%29-f%28n%29+=+%283%5E%28n%2B1%29%2B7%5E%28n%2B1%29-2%29-%283%5En%2B7%5En-2%29\"\r\n" );
document.write( "\"f%28n%2B1%29-f%28n%29+=+3%5E%28n%2B1%29%2B7%5E%28n%2B1%29-2-3%5En-7%5En%2B2%29\"\r\n" );
document.write( "\"f%28n%2B1%29-f%28n%29+=+3%5E%28n%2B1%29-3%5En%2B7%5E%28n%2B1%29-7%5En\"}\r\n" );
document.write( "\"f%28n%2B1%29-f%28n%29+=+3%5En%2A%283-1%29%2B7%5En%2A%287-1%29\"\r\n" );
document.write( "\"f%28n%2B1%29-f%28n%29+=+3%5En%2A%282%29%2B7%5En%2A%286%29\"\r\n" );
document.write( "\"f%28n%2B1%29-f%28n%29+=+2%2A3%5En%2B6%2A7%5En\"\r\n" );
document.write( "\"f%28n%2B1%29-f%28n%29+=+2%283%5En%2B3%2A7%5En%29\"\r\n" );
document.write( "\"f%28n%2B1%29-f%28n%29+=+2%283%283%5E%28n-1%29%2B7%5En%29%5E%22%22%29\"\r\n" );
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document.write( "But we must now show that \r\n" );
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document.write( "\"g%28n%29=3%5E%28n-1%29%2B7%5En\" is divisible by 4 \r\n" );
document.write( "\"g%281%29=3%5E%281-1%29%2B7%5E1=3%5E0%2B7=1%2B7=8=4%2A2\" is divisible by 4\r\n" );
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document.write( "As above, we only need to show that the difference g(n+1)-g(n) is divisible by\r\n" );
document.write( "4. For if that is divisible by 4 then the assumption that g(n) is divisible\r\n" );
document.write( "by 4 will show that g(n+1) is divisible by 4 because the sum of two\r\n" );
document.write( "multiples of 4 is a multiple of 4.\r\n" );
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document.write( "\"g%28n%2B1%29-g%28n%29=3%5E%28n%2B1-1%29%2B7%5E%28n-1%29-3%5E%28n-1%29-7%5En\"\r\n" );
document.write( "\"g%28n%2B1%29-g%28n%29=3%5En%2B7%5E%28n-1%29-3%5E%28n-1%29-7%5En\"\r\n" );
document.write( "\"g%28n%2B1%29-g%28n%29=3%5En-3%5E%28n-1%29%2B7%5E%28n-1%29-7%5En\"\r\n" );
document.write( "\"g%28n%2B1%29-g%28n%29=3%5E%28n-1%29%2A%283-1%29%2B7%5E%28n-1%29%281-7%29\"\r\n" );
document.write( "\"g%28n%2B1%29-g%28n%29=3%5E%28n-1%29%2A%282%29%2B7%5E%28n-1%29%28-6%29\"\r\n" );
document.write( "\"g%28n%2B1%29-g%28n%29=2%2A3%5E%28n-1%29-6%2A7%5E%28n-1%29\"\r\n" );
document.write( "\"g%28n%2B1%29-g%28n%29=2%283%5E%28n-1%29-3%2A7%5E%28n-1%29%29\"\r\n" );
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document.write( "The expression in parentheses is even, because\r\n" );
document.write( "all products and powers of odd numbers are odd numbers,\r\n" );
document.write( "and the difference of two odd numbers is an even number.\r\n" );
document.write( "Therefore \"g%28n%2B1%29-g%28n%29\" is a multiple of 4.\r\n" );
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document.write( "Therefore \"f%28n%2B1%29-f%28n%29\" is a multiple of 8.\r\n" );
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document.write( "Edwin
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