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2018 AMC 12B Problem 9

Problem 9 of 25EasierAlgebra

What is ∑i=1100∑j=1100(i+j)? \sum_{i=1}^{100}\sum_{j=1}^{100}(i+j)?

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Solution

Splitting the sum, ∑i=1100∑j=1100(i+j)=∑i=1100∑j=1100i+∑i=1100∑j=1100j=100∑i=1100i+100∑j=1100j. \begin{gathered} \sum_{i=1}^{100}\sum_{j=1}^{100}(i+j) \\ =\sum_{i=1}^{100}\sum_{j=1}^{100}i \\ {}+\sum_{i=1}^{100}\sum_{j=1}^{100}j \\ =100\sum_{i=1}^{100}i \\ {}+100\sum_{j=1}^{100}j. \end{gathered} Since ∑k=1100k=5050,\sum_{k=1}^{100}k=5050, this equals 100⋅5050100\cdot5050 +100⋅5050+100\cdot5050 =1,010,000.=1{,}010{,}000. Thus, the correct answer is E.

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Concepts: summation · arithmetic sequence

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.