document.write( "Question 170325: Use mathematical induction to prove: 4^n+1 + 5^2n-1 is divisible by 21 \n" ); document.write( "
Algebra.Com's Answer #125806 by Edwin McCravy(20056)\"\" \"About 
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Use mathematical induction to prove: \"4%5E%28n%2B1%29+%2B+5%5E%282n-1%29\" is divisible by 21
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document.write( "Let \"f%28n%29=4%5E%28n%2B1%29+%2B+5%5E%282n-1%29\"\r\n" );
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document.write( "First prove that there is at least one\r\n" );
document.write( "value of n for which \"f%28n%29\" is divisible by 21:\r\n" );
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document.write( "\"4%5E%28n%2B1%29+%2B+5%5E%282n-1%29\"\r\n" );
document.write( "\"4%5E%281%2B1%29+%2B+5%5E%282%2A1-1%29\"\r\n" );
document.write( "\"4%5E%282%29+%2B+5%5E%282-1%29\"\r\n" );
document.write( "\"16+%2B+5%5E%281%29\"\r\n" );
document.write( "\"16%2B5\"\r\n" );
document.write( "\"21\"\r\n" );
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document.write( "Strategy:\r\n" );
document.write( "If n=k is a value of n so that f(n=k) is divisible by 21,\r\n" );
document.write( "then if f(k+1) and f(k) differ by a multiple of 21, then \r\n" );
document.write( "f(k+1) will also be divisible by 21.\r\n" );
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document.write( "So, we will consider the difference f(k+1)-f(k)  \r\n" );
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document.write( "But first we must calculate f(n=k+1):\r\n" );
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document.write( "\"f%28n%29=4%5E%28n%2B1%29+%2B+5%5E%282n-1%29\"\r\n" );
document.write( "\"f%28k%2B1%29=4%5E%28%28k%2B1%29%2B1%29+%2B+5%5E%282%28k%2B1%29-1%29\"\r\n" );
document.write( "\"f%28k%2B1%29=4%5E%28k%2B1%2B1%29%2B5%5E%282k%2B2-1%29\"\r\n" );
document.write( "\"f%28k%2B1%29=4%5E%28k%2B2%29%2B5%5E%282k%2B1%29\"\r\n" );
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document.write( "Now we consider \r\n" );
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document.write( "And since we know that since f(1) is divisible by 21\r\n" );
document.write( "then f(2) is also divisible by 21, and thus f(3) is\r\n" );
document.write( "also divisible by 21, etc., etc.\r\n" );
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document.write( "Edwin
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