# Write the numbers from 1 to 100 with either an add or subtract sign between each. Can you make the total 2000?

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Write the numbers from 1 to 100 with either an add or subtract sign between each. Can you make the total 2000?
posted Aug 23, 2017

The sum of all the numbers is 5050. Changing the sign of a number reduces the sum by twice that number. Let’s do a little trial and error. Changing the sign of the first k numbers reduces the total by k(k+1). We would like this to be 3050. Solving gives k = \frac{-1 \pm \sqrt{12201}}2 which is about 54. If k = 54 then the sum becomes 5050 - 54\times55 = 2080. We need to reduce the sum by another 80. If we try k = 55 the sum reduces by another 110 and we need to increase the sum by 30. We can do this by changing the sign of 15 again.

So put negative signs before the numbers from 1 to 14, add 15, subtract numbers from 16 to 55 and add the numbers from 56 to 100.

But there are many alternative solutions. The sum of the numbers from 1 to 5 is 15, so you could add the numbers from 1 to 5, subtract the numbers from 6 to 55 then add the numbers from 56 to 100.

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answer Feb 8, 2018

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