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You can put this solution on YOUR website!
You might be looking for a formula that will add up the consecutive natural numbers from 1 to n. You know that f(1) = 1, f(2) = 1 + 2 = 3; f(3) = 1 + 2 + 3 = 6, etc... and ultimatley, f(n) = 1 + 2 + 3 + .... + n. But you don't want to keep on adding the consecutive terms because it'd take you forever, especially if n were big enough.\r
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\n" ); document.write( "\n" ); document.write( "I've seen this one before and the derivation is quite clever. Look again at the expression that you wrote 1 + 2 + 3 + .... + 498 + 499 + 500. Imagine that each number were on a strip of long paper. Fold the paper exactly in half. This way, the 1 will be paired with the 500, the 2 will be paired with the 499, the 3 will be paired with the 498, and so on.\r
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\n" ); document.write( "\n" ); document.write( "If you notice, each pair after the fold ALWAYS has a sum of 501. Aha! But how many pairs would there be? There would be 250 pairs because you folded the strip in half. So 1 + 2 + 3 + 4 + ... + 500 = 501 * 250 = 125250.\r
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\n" ); document.write( "\n" ); document.write( "Working with our example here, N = 500, the last number in our summation. When we \"paired the opposite ends\", we found that the sums were 501, which really is N + 1. Each pair's sum was 501, but how many pairs were there? 250 pairs, which really is N/2.\r
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\n" ); document.write( "\n" ); document.write( "Now, we can put together the formula