document.write( "Question 1035751: Identify the correct proof by induction or a counterexample to disprove the following statement.
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Algebra.Com's Answer #650419 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
\"1%5E3%2B2%5E3%2B3%5E3%2B%22...%22%2Bn%5E3\"\"%22%22=%22%22\"\"%28n%5E2%28n%2B1%29%5E2%29%2F4\"
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document.write( "induction proof:\r\n" );
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document.write( "First let's see what Pk+1 would be:\r\n" );
document.write( "[That's always the first thing to do.  Before you start an induction\r\n" );
document.write( "proof,  you should calculate Pk+1 to see where you're headed]:\r\n" );
document.write( "To do that, replace n by k+1 in \"n%5E2%28n%2B1%29%5E2%2F4\" to see what Pk+1 is, \r\n" );
document.write( "for that is what we are going for, and if we have that beforehand,\r\n" );
document.write( "we'll know when we have arrived and the proof is finished.\r\n" );
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document.write( "Substituting k+1 for n in \"%28n%5E2%28n%2B1%29%5E2%29%2F4\", we have\r\n" );
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document.write( "\"%28%28k%2B1%29%5E2%28%28k%2B1%29%5E%22%22%2B1%29%5E2%29%2F4\" or \r\n" );
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document.write( "\"%28%28k%2B1%29%5E2%28k%2B1%2B1%29%5E2%29%2F4\" or\r\n" );
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document.write( "\"%28%28k%2B1%29%5E2%28k%2B2%29%5E2%29%2F4\"\r\n" );
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document.write( "Now that we know what Pk+1 is, we know where we're going, \r\n" );
document.write( "and we'll know we have arrived if and when we get \"%28%28k%2B1%29%5E2%28k%2B2%29%5E2%29%2F4\". \r\n" );
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document.write( "So now we can start the proof:\r\n" );
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document.write( "P1:  substitute n=1, \r\n" );
document.write( "\"1%5E3\"\"%22%22=%22%22\"\"%281%5E2%281%2B1%29%5E2%29%2F4\"\"%22%22=%22%22\"\"%281%282%29%5E2%29%2F4\"\"%22%22=%22%22\"\"4%2F4\"\"%22%22=%22%22\"\"1\",\r\n" );
document.write( "which is true.\r\n" );
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document.write( "Assume Pk: \"1%5E3%2B2%5E3%2B3%5E3%2B%22...%22%2Bk%5E3\"\"%22%22=%22%22\"\"%28k%5E2%28k%2B1%29%5E2%29%2F4\"\r\n" );
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document.write( "Add (k+1)³ to both sides:\r\n" );
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document.write( "\"1%5E3%2B2%5E3%2B3%5E3%2B%22...%22%2Bk%5E3%2Bred%28%28k%2B1%29%5E3%29\"\"%22%22=%22%22\"\"%28k%5E2%28k%2B1%29%5E2%29%2F4%2Bred%28%28k%2B1%29%5E3%29\"\r\n" );
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document.write( "Get an LCD of 4\r\n" );
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document.write( "\"1%5E3%2B2%5E3%2B3%5E3%2B%22...%22%2Bk%5E3%2B%28k%2B1%29%5E3\"\"%22%22=%22%22\"\"%28k%5E2%28k%2B1%29%5E2%29%2F4%2B4%28k%2B1%29%5E3%2F4\"\r\n" );
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document.write( "\"1%5E3%2B2%5E3%2B3%5E3%2B%22...%22%2Bk%5E3%2B%28k%2B1%29%5E3\"\"%22%22=%22%22\"\"%28k%5E2%28k%2B1%29%5E2%2B4%28k%2B1%29%5E3%29%2F4\"\r\n" );
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document.write( "Factor out (k+1)² in the numerator on the right:\r\n" );
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document.write( "\"1%5E3%2B2%5E3%2B3%5E3%2B%22...%22%2Bk%5E3%2B%28k%2B1%29%5E3\"\"%22%22=%22%22\"\"%28%28k%2B1%29%5E2%28k%5E2%2B4%28k%2B1%29%29%29%2F4\" \r\n" );
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document.write( "Distribute on right:\r\n" );
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document.write( "\"1%5E3%2B2%5E3%2B3%5E3%2B%22...%22%2Bk%5E3%2B%28k%2B1%29%5E3\"\"%22%22=%22%22\"\"%28%28k%2B1%29%5E2%28k%5E2%2B4k%2B4%29%29%2F4\"\r\n" );
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document.write( "Factor on right:\r\n" );
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document.write( "\"1%5E3%2B2%5E3%2B3%5E3%2B%22...%22%2Bk%5E3%2B%28k%2B1%29%5E3\"\"%22%22=%22%22\"\"%28%28k%2B1%29%5E2%28k%5E2%2B4k%2B4%29%29%2F4\"\r\n" );
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document.write( "\"1%5E3%2B2%5E3%2B3%5E3%2B%22...%22%2Bk%5E3%2B%28k%2B1%29%5E3\"\"%22%22=%22%22\"\"%28%28k%2B1%29%5E2%28k%2B2%29%5E2%29%2F4\"\r\n" );
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document.write( "and now we see that we have gotten Pk+1 that we found in the \r\n" );
document.write( "beginning that we were going for.\r\n" );
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document.write( "So the proof is finished.\r\n" );
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document.write( "So since P1 is true, P1 proves P2, P2 proves P3, P3 proves P4,\r\n" );
document.write( "etc., etc., ad infinitum.\r\n" );
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document.write( "Edwin
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