document.write( "Question 909567: Prove by mathematical induction that:\r
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Algebra.Com's Answer #551926 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "What we are to prove is that for any positive integer n,\r\n" );
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document.write( "\"2%5E%282n%29-1\" is divisible by 3, that is, it is a multiple of 3.\r\n" );
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document.write( "For n=1\r\n" );
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document.write( "\"2%5E%282%281%29%29-1=2%5E2-1+=4-1=3\". 3 is a multiple of 3.\r\n" );
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document.write( "So it's true for n=1 \r\n" );
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document.write( "We must prove that if you know it's true for integer n=k, \r\n" );
document.write( "then it will be true for n=k+1.\r\n" );
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document.write( "Assume it's true for n=k.\r\n" );
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document.write( "\"2%5E%282k%29-1=3m\" for some positive integer m\r\n" );
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document.write( "Multiply both sides by \"2%5E2\"\r\n" );
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document.write( "\"2%5E2%2A2%5E%282k%29-2%5E2=2%5E2%2A3m\" for some positive integer m\r\n" );
document.write( "\"2%5E2%2A2%5E%282k%29-4=4%2A3m\" for some positive integer m\r\n" );
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document.write( "\"2%5E%282k%2B2%29-4=12m\"\r\n" );
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document.write( "Write -4 as -1-3\r\n" );
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document.write( "\"2%5E%282%28k%2B1%29%29-1-3=12m\"\r\n" );
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document.write( "\"2%5E%282%28k%2B1%29%29-1=12m%2B3\"\r\n" );
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document.write( "\"2%5E%282%28k%2B1%29%29-1=3%284m%2B1%29%7D%7D%2C+thus+%7B%7B%7B2%5E%282%28k%2B1%29%29-1\" is a\r\n" );
document.write( "multiple of 3.\r\n" );
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document.write( "Therefore since the theorem is true for n=k=1, it's true for n=k+1=2.\r\n" );
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document.write( "And since the theorem is true for n=k=2, it's true for n=k+1=3\r\n" );
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document.write( "Thus since the theorem is true for n=k=3, it's true for n=k+1=4\r\n" );
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document.write( "And this goes on forever, for we've proved it can't stop!\r\n" );
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document.write( "Edwin
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