document.write( "Question 1137739: Prove by induction that for all positive integers value of n:
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Algebra.Com's Answer #761597 by Edwin McCravy(20056) $\"\"$ $\"About$

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Prove by induction that for all positive integers value of n:
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document.write( "First we show that it is true when n=1:\r\n" );
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document.write( ", which is a multiple of 9 since 9∙3 = 27\r\n" );
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document.write( "Now we show that IF it were true when n=k, that it would also be true\r\n" );
document.write( "when n=k+1.\r\n" );
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document.write( "We let a multiple of 9 be 9P where P is an integer such that\r\n" );
document.write( "when n=k,  equals to 9P.  That is, we examine\r\n" );
document.write( "what would happen IF this were true for some value of k:\r\n" );
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document.write( "we hope (but do not know!) that if that were true, then when we substitute\r\n" );
document.write( "(k+1) for n, like this:\r\n" );
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document.write( " that it will be a multiple of 9 also, so we simplify it:\r\n" );
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document.write( "We notice that the first term is 25 times the first term of , \r\n" );
document.write( "so we see what we must add to 25 times  to get .\r\n" );
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document.write( "25 times  is . So we write\r\n" );
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document.write( " as\r\n" );
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document.write( " which is\r\n" );
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document.write( ", which is\r\n" );
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document.write( ", which is\r\n" );
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document.write( ", which is\r\n" );
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document.write( ", which is a multiple of 9.\r\n" );
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document.write( "Now since we have a value of n, namely k=1, for which \r\n" );
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document.write( " is a multiple of 9, then we have\r\n" );
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document.write( "proved that we also have another value of n, namely k+1 or 1+1 or\r\n" );
document.write( "2 for which  is a multiple of 9.\r\n" );
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document.write( "Now since we have a value of n, namely k=2, for which \r\n" );
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document.write( " is a multiple of 9, then we have\r\n" );
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document.write( "proved that we also have another value of n, namely k+1 or 2+1 or\r\n" );
document.write( "3 for which  is a multiple of 9.\r\n" );
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document.write( "Now since we have a value of n, namely k=3, for which \r\n" );
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document.write( " is a multiple of 9, then we have\r\n" );
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document.write( "proved that we also have another value of n, namely k+1 or 3+1 or\r\n" );
document.write( "4 for which  is a multiple of 9.\r\n" );
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document.write( "And so on and on for all positive integer values of n.\r\n" );
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document.write( "Edwin

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