document.write( "Question 1168361: (a) Compute the sum:101^2 - 97^2 + 93^2 - 89^2 + ...+ 5^2 - 1^2.
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\n" ); document.write( "Compute the sum (a +(2n+1)d)^2- (a + (2n)d)^2 +(a + (2n-1)d)^2 - (a+(2n-2)d)^2 + ... + (a+d)^2 - a^2.
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Algebra.Com's Answer #792953 by ikleyn(52794) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Compute the sum: 101^2 - 97^2 + 93^2 - 89^2 + ...+ 5^2 - 1^2.
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document.write( "SUM = 101^2 - 97^2 + 93^2 - 89^2 + ...+ 5^2 - 1^2 = group the term in pairs =\r\n" );
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document.write( "    101^2 - 97^2 +\r\n" );
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document.write( "  +  93^2 - 89^2 + \r\n" );
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document.write( "  +  85^2 - 81^2 +\r\n" );
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document.write( "  . . . . . . . . . . .\r\n" );
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document.write( "  +   5^2 - 1^2.\r\n" );
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document.write( "To each pair (= to each short line) apply the identity  a^2 - b^2 = (a+b)*(a-b).\r\n" );
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document.write( "Notice that the second factor in the right side of this identity is always equal to 4.\r\n" );
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document.write( "Therefore, you get\r\n" );
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document.write( "SUM = 4*((101+97) + (93+89) + (85+81) + . . . + (5+1)) = \r\n" );
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document.write( "    = 4*( 101+97  +  93+89  +  85+81  + . . . +  5+1 )\r\n" );
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document.write( "The sum in parentheses is the sum of the arithmetic progression with the first term of 1, the common difference of 4\r\n" );
document.write( "and the last term of 101.\r\n" );
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document.write( "The number of terms in this progression is   = 26.\r\n" );
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document.write( "Therefore the sum in parentheses is   = 1326.\r\n" );
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document.write( "Hence, the original sum is  SUM = 4*1326 = 5304.\r\n" );
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document.write( "ANSWER.  The requested sum is  5304.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Regarding the other problem, I think that the same idea works there, too.\r
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