document.write( "Question 40302: Hello!
\n" ); document.write( "Here's a tutor who is a bit rusty on this subject. Is there a simple expression for the following sum?\r
\n" ); document.write( "\n" ); document.write( "\"sum%28i%5E2%2Ci=a%2Ca%2B%28n-1%29%29\"\r
\n" ); document.write( "\n" ); document.write( "For example,
\n" ); document.write( "1 + 4 + 9 + 16 + ... + 100
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\n" ); document.write( "25 + 36 + 49
\n" ); document.write( "etc.\r
\n" ); document.write( "\n" ); document.write( "Thank you very much! \n" ); document.write( "
Algebra.Com's Answer #25705 by AnlytcPhil(1807)\"\" \"About 
You can put this solution on YOUR website!
I'm rusty too, so I'll have to derive it from scratch.\r\n" );
document.write( "First we'll find a formula for \r\n" );
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document.write( "S(k) = \"sum%28i%5E2%2Ci=1%2Ck%29\"\r\n" );
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document.write( "then your summation is\r\n" );
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document.write( "S(a+n-1) - S(a-1)\r\n" );
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document.write( "{S(k)} = 1², 1²+2², 1²+2²+3², ··· , 1²+2²+3²+···+k²\r\n" );
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document.write( "Write out some of the terms\r\n" );
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document.write( "1, 5, 14, 30, 55, 91, 140\r\n" );
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document.write( "Make a difference table, i.e, list the terms\r\n" );
document.write( "vertically in a column placing the difference \r\n" );
document.write( "between each pair of successive terms between \r\n" );
document.write( "them to the right.  Then do the same to the \r\n" );
document.write( "second column, until you get a column which all\r\n" );
document.write( "contain the same number:\r\n" );
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document.write( "  1\r\n" );
document.write( "     4\r\n" );
document.write( "  5     5\r\n" );
document.write( "     9     2\r\n" );
document.write( " 14     7 \r\n" );
document.write( "    16     2\r\n" );
document.write( " 30     9\r\n" );
document.write( "    25     2\r\n" );
document.write( " 55    11 \r\n" );
document.write( "    36     2\r\n" );
document.write( " 91    13\r\n" );
document.write( "    49\r\n" );
document.write( "140\r\n" );
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document.write( "It took three difference columns to get a column of\r\n" );
document.write( "all 2's, so we will see if a third degree polynomial\r\n" );
document.write( "in k is possible for the formula:\r\n" );
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document.write( "S(k) = Ak³ + Bk² + Ck + D\r\n" );
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document.write( "Then substituting  \r\n" );
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document.write( " 1 = A(1)³ + B(1)² + C(1) + D\r\n" );
document.write( " 5 = A(2)³ + B(2)² + C(2) + D\r\n" );
document.write( "14 = A(3)³ + B(3)² + C(3) + D\r\n" );
document.write( "30 = A(4)³ + B(4)² + C(4) + D\r\n" );
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document.write( "Giving us this system of 4 equations in 4 unknowns\r\n" );
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document.write( "  A +   B +  C + D = 1\r\n" );
document.write( " 8A +  4B + 2C + D = 5 \r\n" );
document.write( "27A +  9B + 3C + D = 14\r\n" );
document.write( "64A + 16B + 4C + D = 30\r\n" );
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document.write( "Solve this system and we get\r\n" );
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document.write( "A = 1/3, B = 1/2, C = 1/6, D = 0\r\n" );
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document.write( "So if we are right our formula is\r\n" );
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document.write( "S(k) = (1/3)k³ + (1/2)k² + (1/6)k\r\n" );
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document.write( "S(k) = (2k³ + 3k² + k)/6\r\n" );
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document.write( "S(k) = k(2k² + 3k + 1)/6\r\n" );
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document.write( "S(k) = k(k+1)(2k+1)/6 \r\n" );
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document.write( "This can be proved correct by induction.\r\n" );
document.write( "It is true for k=1, so if we add (k+1)²\r\n" );
document.write( "to both sides\r\n" );
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document.write( "S(k) + (k+1)² = k(k+1)(2k+1)/6 + (k+1)²\r\n" );
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document.write( "= [(k+1)/6][k(2k+1) + 6(k+1)]\r\n" );
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document.write( "= [(k+1)/6][2k²+7k+6] = [(k+1)/6][(k+2)(2k+3)]\r\n" );
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document.write( "= (k+1)(k+2)(2k+3)/6 which equals S(k+1)\r\n" );
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document.write( "so we have the right formula.\r\n" );
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document.write( "Now your problem is to find \r\n" );
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document.write( "S(a+n-1) - S(a-1)\r\n" );
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document.write( "(a+n-1)(a+n-1+1)(2(a+n-1)+1)/6 -\r\n" );
document.write( "                      (a-1)(a-1+1)(2(a-1)+1)/6\r\n" );
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document.write( "(a+n-1)(a+n)(2a+2n-1)/6 - (a-1)(a)(2a-1)/6 =\r\n" );
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document.write( "[(a+n-1)(a+n)(2a+2n-1) - a(a-1)(2a-1)]/6\r\n" );
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document.write( "I'll let you simplify that if you like.\r\n" );
document.write( "I'm too tired! :-)\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com

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