document.write( "Question 1025935: Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false.
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Algebra.Com's Answer #641207 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ... + 4n( 4n + 2) = 4(4n+1)(8n+7)/6\r\n" );
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document.write( "is false because it isn't even true when n=1.\r\n" );
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document.write( "4 ⋅ 6  = 4(4*1+1)(8*1+7)/6\r\n" );
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document.write( "   24 = 4(5)(15)/6\r\n" );
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document.write( "   24 = 50\r\n" );
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document.write( "Also 4n(4n+2) is not even the correct formula\r\n" );
document.write( "for the nth term.  The correct nth term formula\r\n" );
document.write( "of 4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ...\r\n" );
document.write( "is (n+3)(n+5).  So it's wrong all the way around.\r\n" );
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document.write( " 1²+4²+7²+∙∙∙+(3n-2)² = n(6n²-3n-1)/2\r\n" );
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document.write( "That appears to be correct.\r\n" );
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document.write( "If n=1 then  1² = 1(6∙1²-3∙1-1)/2 = 1(6-3-1)/2 = 1(2)/2 = 1 which \r\n" );
document.write( "shows that the proposition is true when n=1.\r\n" );
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document.write( "Let us assume that n=k is some integer (perhaps 1) such that\r\n" );
document.write( "1²+4²+7²+∙∙∙+(3k-2)² = k(6k²-3k-1)/2.  \r\n" );
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document.write( "We want to show that the expression n(6n²-3n-1)/2 with k+1 substituted\r\n" );
document.write( "for n also holds for the sum of the first k+1 terms.  That is, we want \r\n" );
document.write( "to be able to erase the question mark here: \r\n" );
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document.write( " 1²+4²+7²+∙∙∙+[3(k+1)-2]² ≟ (k+1)[6(k+1)²-3(k+1)-1]/2 \r\n" );
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document.write( "or upon multiplying out the right side (you do that)\r\n" );
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document.write( " 1²+4²+7²+∙∙∙+[3(k+1)-2]² ≟ 3k³+15k²/2+11k/2+1\r\n" );
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document.write( "We start with our assumption:\r\n" );
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document.write( "1²+4²+7²+∙∙∙+(3k-2)² = k(6k²-3k-1)/2.\r\n" );
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document.write( "or\r\n" );
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document.write( "1²+4²+7²+∙∙∙+(3k-2)² = 3k³-3k²/2-k/2\r\n" );
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document.write( "and add [3(k+1)-2]² to both sides:\r\n" );
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document.write( "1²+4²+7²+∙∙∙+(3k-2)²+[3(k+1)-2]² = k(6k²-3k-1)/2 + [3(k+1)-2]²\r\n" );
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document.write( "Multiplying the right side all the way out, (you do that) \r\n" );
document.write( "we get \r\n" );
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document.write( "1²+4²+7²+∙∙∙+(3k-2)²+[3(k+1)-2]² = 3k³+(15/2)k²+(11/2)k+1\r\n" );
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document.write( "and that is the same right side as the right side\r\n" );
document.write( "of above when we had the equal sign with the question\r\n" );
document.write( "mark above it.\r\n" );
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document.write( "1²+4²+7²+∙∙∙+[3(k+1)-2]² ≟ 3k³+(15/2)k²+(11/2)k+1 \r\n" );
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document.write( "Edwin

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