document.write( "Question 878891: show that 1/2!+2/3!+3/4!+...+n/(n+1)!=1-1/(n+1)!. where n!=1*2*3*...*n.\r
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Algebra.Com's Answer #535667 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"1%2F2%21%2B2%2F3%21%2B3%2F4%21%2B%22%22%2A%22%22%2A%22%22%2A%22%22%2Bn%2F%28n%2B1%29%21\"\"%22%22=%22%22\"\"1-1%2F%28n%2B1%29%21\"\r\n" );
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document.write( "First we prove that when n=1, and there is just 1 term on the left,\r\n" );
document.write( "the the above is true:\r\n" );
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document.write( "\"1%2F2%21\"\"%22%22=%22%22\"\"1%2F%281%2B1%29%21\"\r\n" );
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document.write( "That is true since 1+1=2\r\n" );
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document.write( "Now since we have a value of n where it works, we can assume that\r\n" );
document.write( "there is at least one value of n=k where it works, for we know it\r\n" );
document.write( "works for k=1 as we just showed.\r\n" );
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document.write( "So we can be sure that there is at least one value of k where\r\n" );
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document.write( "\"1%2F2%21%2B2%2F3%21%2B3%2F4%21%2B%22%22%2A%22%22%2A%22%22%2A%22%22%2Bk%2F%28k%2B1%29%21\"\"%22%22=%22%22\"\"1-1%2F%28k%2B1%29%21\"\r\n" );
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document.write( "is true, even if it were only true only for k=1.\r\n" );
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document.write( "Now we will add the next term \"red%28%28k%2B1%29%2F%28k%2B2%29%21%29\" to both sides:\r\n" );
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document.write( "\"%22%22=%22%22\"\"1-1%2F%28k%2B1%29%21%2Bred%28%28k%2B1%29%2F%28k%2B2%29%21%29%29\"\r\n" );
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document.write( "                              \"%22%22=%22%22\"\"1-1%2F%28k%2B1%29%21%2Bred%28%28k%2B1%29%2F%28k%2B2%29%28k%2B1%29%21%29%29\"\r\n" );
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document.write( "                              \"%22%22=%22%22\"\r\n" );
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document.write( "                              \"%22%22=%22%22\"\r\n" );
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document.write( "                              \"%22%22=%22%22\"\"1-%28k%2B2%29%2F%28k%2B2%29%21%2B%28k%2B1%29%2F%28k%2B2%29%21%29\"\r\n" );
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document.write( "Notice there is a \"-\" before the \"%28k%2B2%29%2F%28k%2B2%29%21\"\r\n" );
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document.write( "                              \"%22%22=%22%22\"\"1%2B%28-k-2%29%2F%28k%2B2%29%21%2B%28k%2B1%29%2F%28k%2B2%29%21%29\"\r\n" );
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document.write( "                              \"%22%22=%22%22\"\"1%2B%28-k-2%2Bk%2B1%29%2F%28k%2B2%29%21\"\r\n" );
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document.write( "                              \"%22%22=%22%22\"\"1%2B%28-1%29%2F%28k%2B2%29%21\"\r\n" );
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document.write( "\"%22%22=%22%22\"\"1-1%2F%28k%2B2%29%21\"\r\n" );
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document.write( "That's the formula with n equaling to k+1\r\n" );
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document.write( "So we have shown that if the formula works for some n=k, then it works \r\n" );
document.write( "also for n=k+1\r\n" );
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document.write( "Therefore since we have shown that it works for n=k=1, this proves that it\r\n" );
document.write( "also works for n=k=2.\r\n" );
document.write( "Therefore since we have shown that it works for n=k=2, this proves that it\r\n" );
document.write( "also works for n=k=3.\r\n" );
document.write( "Therefore since we have shown that it works for n=k=3, this proves that it\r\n" );
document.write( "also works for n=k=4.\r\n" );
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document.write( "etc. etc.\r\n" );
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document.write( "So it works for ALL values of n\r\n" );
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document.write( "Edwin
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