document.write( "Question 1046622: I am stuck on Guass's Approach with odd numbers. The problem is 1+3+5+7+...+999. I've looks for examples everywhere. I hope you can help!
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Algebra.Com's Answer #662066 by Theo(13342) $\"\"$ $\"About$

You can put this solution on YOUR website!
gauss's trick is described here.\r
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\n" ); document.write( "\n" ); document.write( "https://nzmaths.co.nz/gauss-trick-staff-seminar\r
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\n" ); document.write( "\n" ); document.write( "i used his method to determine the value of the sum of the odd number from 1 to 999.\r
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\n" ); document.write( "\n" ); document.write( "the sum of the number from 1 to 1000 can be found as follows:\r
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\n" ); document.write( "\n" ); document.write( "in normal order, the sequence is 1,2,3,4,5.
\n" ); document.write( "in reverse order, the sequence is 1000,999,998,997,994
\n" ); document.write( "pair the normal order terms with the reverse order terms and you get:
\n" ); document.write( "1 + 1000 = 1001
\n" ); document.write( "2 + 999 = 1001
\n" ); document.write( "3 + 998 = 1001
\n" ); document.write( "etc.
\n" ); document.write( "there are 1000 of such pairings.
\n" ); document.write( "1000 * 1001 = 1001000.
\n" ); document.write( "that's double the sum of 1 to 1000.
\n" ); document.write( "divide that by 2 and you get the sum of the numbers from 1 to 1000 = 500500.\r
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\n" ); document.write( "\n" ); document.write( "there are 500 pairs of numbers from 1 to 1000.
\n" ); document.write( "the pairs are:
\n" ); document.write( "1,2
\n" ); document.write( "3,4
\n" ); document.write( "5,6
\n" ); document.write( "etc.
\n" ); document.write( "in each pair, the even number is 1 more than the odd number.\r
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\n" ); document.write( "\n" ); document.write( "since there are 500 pairs, then the sum of the even numbers will be 500 more than the sum of the odd numbers.\r
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\n" ); document.write( "\n" ); document.write( "let x = the sum of the odd number.
\n" ); document.write( "x + 500 is the sum of the even numbers.\r
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\n" ); document.write( "\n" ); document.write( "you get x + x + 500 = 500500
\n" ); document.write( "simplify to get 2x + 500 = 500500
\n" ); document.write( "subtract 500 from 500500 to get:
\n" ); document.write( "2x = 500000
\n" ); document.write( "divide both sides by 2 to get:
\n" ); document.write( "x = 250000
\n" ); document.write( "that's the sum of the odd numbers.
\n" ); document.write( "the sum of the even numbers is 250500
\n" ); document.write( "add them up and you get the sum of the number is 500500.\r
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\n" ); document.write( "\n" ); document.write( "i check with excel and got those tallies, so the formula works.\r
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\n" ); document.write( "\n" ); document.write( "the sum of the odd number is 250,000.
\n" ); document.write( "the sum of the even numbers is 250,500.\r
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\n" ); document.write( "\n" ); document.write( "here's a picture of my excel analysis.\r
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