document.write( "Question 205420: If the sum of the first 100 integers: 1+2+3...+99+100=5050, then the sum of the 50 odd integers: 1+3+5...+99=? \n" ); document.write( "
Algebra.Com's Answer #155060 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let the sum of the first fifty odd numbers be N\r\n" );
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document.write( "  1  +   3   +   5   + ··· +  95  +  97  +  99  =  N\r\n" );
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document.write( "Write the above equation down, then right under them,\r\n" );
document.write( "write the same numbers added in the reverse order. \r\n" );
document.write( "Like this:\r\n" );
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document.write( "  1  +   3   +   5   + ··· +  95  +  97  +  99  =  N\r\n" );
document.write( " 99  +  97   +  95   + ··· +   5  +   3  +   1  =  N\r\n" );
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document.write( "Now add those two equations term by term, like this:\r\n" );
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document.write( "  1  +   3   +   5   + ··· +  95  +  97  +  99  =  N\r\n" );
document.write( " 99  +  97   +  95   + ··· +   5  +   3  +   1  =  N\r\n" );
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document.write( "100  + 100   + 100   + ··· + 100  + 100  + 100  = 2N\r\n" );
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document.write( "There are fifty terms, all 100's in the bottom equation,\r\n" );
document.write( "so obviously that's 50x100 or 5000, so the above becomes:\r\n" );
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document.write( "                                           5000 = 2N\r\n" );
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document.write( "Divide both sides by 2                     2500 = N\r\n" );
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document.write( "So the sum is 2500.\r\n" );
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document.write( "(You didn't need the sum of the first 100 positive integers.)\r\n" );
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document.write( "Edwin
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