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# Determine the sum for the arithmetic series?

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- BrianLv 78 years ago
sum(m=10 to 20)(30 - m) =

sum(m=10 to 20)(30) - sum(m=10 to 20)(m) =

30*sum(m=10 to 20)(1) - [sum(m=1 to 20)(m) - sum(m=1 to 9)(m)] =

30*(20 - 10 + 1) - [(20*(20 + 1)/2) - (9*(9 + 1)/2)] =

30*11 - [210 - 45] = 330 - 165 = 165.

Edit: Note that in general sum(m=1 to n)(m) = n*(n + 1)/2, and that

sum(m=n to p)(1) = p - n + 1.

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