You can put this solution on YOUR website! The formula to use is:
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For this problem N is the number of terms and N = 1000, A is the first term which is 1, and L is the last term which is 1000.
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So the equation becomes:
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And this simplifies to:
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Multiplying out the numerator gives:
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And finally dividing by 2 results in:
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Another way to view this problem is to write the series for which you want the sum in ascending order from left to right:
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1 + 2 + 3 + 4 + .............. + 997 + 998 + 999 + 1000
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Now directly under it, write the same series but in backward order. So the two series together look like:
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1 + 2 + 3 + 4 + .............. + 997 + 998 + 999 + 1000
1000+999+998 +997 ............... +4 + 3 + 2 + 1
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The vertical typing alignment is a little bit of a problem here. But picture this: instead of adding the two series from left to right, let's add them vertically (in vertical columns). Notice that each vertical addition results in the same answer of 1001. The first vertical addition is (1 + 1000). The second is (2 + 999). The third is (3 + 998). The fourth is (4 + 997) and so on until at the very right end you have (997 + 4) then (998 + 3) then (999 + 2) and finally (1000 + 1). So you have 1000 vertical additions and each one of them results in the sum of 1001. If you want to add all those vertical sums together you can do that by just multiplying 1000 times 1001 and you get 1001000. But we added two of the series that we are interested (one forward and one backward) so we have found twice the answer that we need. Therefore, if we divide the 1001000 by 2 we get the sum of just one of the series. Hopefully that makes some degree of sense and you can see why the formula is what it is.
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