document.write( "Question 1080118: How to prove \"n%5E2-n%2B14\" is divisible by 2 using mathematical induction given it's true for all natural numbers (n)? \n" ); document.write( "
Algebra.Com's Answer #694365 by Edwin McCravy(20055)\"\" \"About 
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document.write( "We are to prove:\r\n" );
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document.write( "Pn:  n²-n+14 is divisible by 2 \r\n" );
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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" );
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document.write( "[That's always the best 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" );
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document.write( "To do that, replace n by k+1 in n²-n+14 to see what the Pk+1 is,\r\n" );
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document.write( "for that is what we are going for, and if we have that beforehand,\r\n" );
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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 n²-n+14, we have\r\n" );
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document.write( "(k+1)²-(k+1)+14 = k²+2k+1-k-1+14 = k²+k+14.\r\n" );
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document.write( "So now we have Pk+1, which is where we'll be headed\r\n" );
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document.write( "Pk+1:  k²+k+14 is divisible by 2 \r\n" );
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document.write( "Now that we know what Pk+1 will have to be, we know where we're going, \r\n" );
document.write( "and we'll know we have arrived if and when we get that. \r\n" );
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document.write( "So now we can start the proof:\r\n" );
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document.write( "P1:  1²-1+14 is divisible by 2\r\n" );
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document.write( "That's true because 1²-1+14 = 1-1+14 = 14, which is divisible by 2 \r\n" );
document.write( "because (2)(7) = 14  \r\n" );
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document.write( "Assume k is some integer for which k is true.  We know there is at least\r\n" );
document.write( "one such value, because we just proved P1.\r\n" );
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document.write( "Pk:  k²-k+14 is divisible by 2 \r\n" );
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document.write( "We look at the expression in Pk+1, that we're going for, and realize that \r\n" );
document.write( "the difference between the expression in Pk+1 and the expression in Pk is\r\n" );
document.write( "(k²+k+14)-(k²-k+14) = k²+k+14-k²+k-14 = 2k \r\n" );
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document.write( "2k is a multiple of 2, and we know that if we add two multiples of 2 we\r\n" );
document.write( "get another multiple of 2.\r\n" );
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document.write( "So assuming \r\n" );
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document.write( "Pk:  k²-k+14 is divisible by 2\r\n" );
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document.write( "     k²-k+14 + 2k is also divisible by 2\r\n" );
document.write( "So\r\n" );
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document.write( "     k²+k+14 is divisible by 2\r\n" );
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document.write( "and that is Pk+1\r\n" );
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document.write( "So the proof is finished.\r\n" );
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document.write( "Now we show that it is finished:\r\n" );
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document.write( "So since P1 is true, 1 is a possible value for k, and therefore\r\n" );
document.write( "P1+1 or P2 is true,  so P1 proves P2, P2 proves P3, P3 proves\r\n" );
document.write( "P4, etc., etc., ad infinitum.\r\n" );
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document.write( "No matter how large an integer we have proved that P is true, \r\n" );
document.write( "we have proved that the next integer will also be one in which P \r\n" );
document.write( "is also true.\r\n" );
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document.write( "Edwin
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