document.write( "Question 696538: Can someone please help with conjectures? I do not understand how to create a formula for these. I get how 1 = 1, or n^2 and #2 with 3^2 but that is as far as I understand.\r
\n" ); document.write( "\n" ); document.write( "Consider the following four equations:
\n" ); document.write( "1) 1 = 1
\n" ); document.write( "2) 2 + 3 + 4 = 1 + 8
\n" ); document.write( "3) 5 + 6 + 7 + 8 + 9 = 8 + 27
\n" ); document.write( "4) 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64 = 91\r
\n" ); document.write( "\n" ); document.write( "Conjecture the general formula suggested by these four equations, and prove your conjecture. \n" ); document.write( "
Algebra.Com's Answer #429105 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( " 1) 1 = 1 \r\n" );
document.write( " 2) 2 + 3 + 4 = 1 + 8 \r\n" );
document.write( " 3) 5 + 6 + 7 + 8 + 9 = 8 + 27 \r\n" );
document.write( " 4) 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64 = 91\r\n" );
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document.write( "Each left side is an arithmetic series\r\n" );
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document.write( "1.  First we make a conjecture about the first term of each \r\n" );
document.write( "arithmetic series.  \r\n" );
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document.write( "Those first terms on the left go 1,2,5,10.  Let's compare\r\n" );
document.write( "that to a list of squares beginning with 0², i.e., 1.2.4.9:\r\n" );
document.write( "n      1,2,3, 4\r\n" );
document.write( "an     1,2,5,10\r\n" );
document.write( "(n-1)² 0,1,4, 9\r\n" );
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document.write( "each number is square of 1 more than the square of 1 less than n\r\n" );
document.write( "so the general term is (n-1)²+1, or n²-2n+1+1or n²-2n+2.\r\n" );
document.write( " n²-2n+2\r\n" );
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document.write( "2.  Next we make a conjecture about the number of terms in each \r\n" );
document.write( "arithmetic series. \r\n" );
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document.write( "The number of terms on the left go 1,3,5,7 which is the odd numbers.\r\n" );
document.write( "We compare them to the even integers, 2n\r\n" );
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document.write( "n      1,2,3,4\r\n" );
document.write( "an     1,3,5,7\r\n" );
document.write( "2n     2,4,6,8\r\n" );
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document.write( "We see that each is one less than 2n. So the general term for the\r\n" );
document.write( "number of terms is 2n-1\r\n" );
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document.write( "3.  Next we make a conjecture about the common difference of the \r\n" );
document.write( "terms of each arithmetic series on the left.  That's easy. The\r\n" );
document.write( "difference is alsways 1.\r\n" );
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document.write( "The formula for the sum of an arithmetic series is\r\n" );
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document.write( "Sn = \"n%2F2\"[2a1 + (n-1)d]\r\n" );
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document.write( "except in place of n on the right side, we use the general expression \r\n" );
document.write( "for the number of term.  So we replace n by 2n-1.  We replace\r\n" );
document.write( "a1 by the general expression for the first term, n²-2n+2,\r\n" );
document.write( "and of course d by 1:   \r\n" );
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document.write( "Sn = \"%282n-1%29%2F2\"[2(n²-2n+2) + ({2n-1}-1)1], and simplify:\r\n" );
document.write( "Sn = \"%282n-1%29%2F2\"[2n²-4n+4 + (2n-1-1)],\r\n" );
document.write( "Sn = \"%282n-1%29%2F2\"[2n²-4n+4+2n-1-1],  \r\n" );
document.write( "Sn = \"%282n-1%29%2F2\"[2n²-2n+2],\r\n" );
document.write( "Sn = \"%282n-1%29%2F2\"[2(n²-n+1)],         \r\n" );
document.write( "Sn = (2n-1)(n²-n+1)\r\n" );
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document.write( "So (2n-1)(n²-n+1) is the general term for the left side.\r\n" );
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document.write( "Now we look at the right sides.  All of them have two terms\r\n" );
document.write( "except the first, but it can be considered as also having\r\n" );
document.write( "two terms 0 + 1.  So we have\r\n" );
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document.write( "0 + 1, 1 + 8, 8 + 27, 27 + 64.  We recognize those as cubes:\r\n" );
document.write( "0³+1³, 1³+2³, 2³+ 3³, 3³ + 8³\r\n" );
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document.write( "So the general term on the right is (n-1)³+n³\r\n" );
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document.write( "We can factor that as the sum of two cubes:\r\n" );
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document.write( "(n-1)³+n³ = [(n-1)+n]{(n-1)²-(n-1)n+n²] = (n-1+n)(n²-2n+1-n²+n+n²)\r\n" );
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document.write( "= (2n-1)(n²-n+1)\r\n" );
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document.write( "That's the same general term we got for the left side. So we have\r\n" );
document.write( "proved the conjecture, that in each case the left side will equal\r\n" );
document.write( "the right side.\r\n" );
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document.write( "Edwin
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