document.write( "Question 1006222: Prove that the sum s of the first n natural numbers is given by the
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Algebra.Com's Answer #622374 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "First we show that the expression \r\n" );
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document.write( "\"expr%28n%2F2%29%28n%2B1%29\"\r\n" );
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document.write( "gives the sum of the first 2 natural numbers:\r\n" );
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document.write( "1+2 = 3\r\n" );
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document.write( "and the expression with n=2 substituted gives:\r\n" );
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document.write( "\"expr%282%2F2%29%282%2B1%29=%281%29%283%29+=+3\"\r\n" );
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document.write( "So the formula holds for n=k=22\r\n" );
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document.write( "Now we know that there is at least one natural number k=2 for which\r\n" );
document.write( "the equation holds for n=k.\r\n" );
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document.write( "Next we show that under the assumption that we just showed, that \r\n" );
document.write( "there exists one natural number n=k for which the equation holds true, \r\n" );
document.write( "then the equation will also hold for n=k+1\r\n" );
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document.write( "Under the assumption that the expression gives the sum of the first\r\n" );
document.write( "n=k natural numbers for some n=k, then\r\n" );
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document.write( "\"expr%28k%2F2%29%28k%2B1%29\"\r\n" );
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document.write( "We add the next natural number (k+1) to the expression:\r\n" );
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document.write( "\"expr%28k%2F2%29%28k%2B1%29%2B%28k%2B1%29\"\r\n" );
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document.write( "We factor out (k+1)\r\n" );
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document.write( "And this equals to the expression \r\n" );
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document.write( "\"expr%28n%2F2%29%28n%2B1%29\" with k+1 substituted for n, since\r\n" );
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document.write( "\"expr%28%28k%2B1%29%2F2%29%28%28k%2B1%29%5E%22%22%2B1%29=expr%28%28k%2B1%29%2F2%29%28k%2B2%29\"\r\n" );
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document.write( "Now since we have shown that it is true when n=k=2, it is therefore\r\n" );
document.write( "true when n=k+1=3.\r\n" );
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document.write( "Now since we have shown that it is true when n=k=3, it is therefore\r\n" );
document.write( "true when n=k+1=4.\r\n" );
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document.write( "Etc., etc., \r\n" );
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document.write( "Therefore there can be no first value of k for which the expression \r\n" );
document.write( "does not hold.  For if there were such first value, the expression\r\n" );
document.write( "would hold for n=k-1 and therefore it would hold for n=k, which would \r\n" );
document.write( "be a contradiction to the assumption that there could be a natural \r\n" );
document.write( "number k for which the expression did not hold.\r\n" );
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document.write( "Edwin
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