What is ∑_i=1^100∑_j=1^100(i+j)?

2018 AMC 12B Problem 9

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What is

\sum_{i=1}^{100}\sum_{j=1}^{100}(i+j)?

100{,}100

500{,}500

505{,}000

1{,}001{,}000

1{,}010{,}000

Split the sum into

\sum\sum i+\sum\sum j

The sum of the first

100

positive integers is

5050

Splitting the sum,

\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

\sum_{k=1}^{100}k=5050,

this equals

100\cdot5050

+100\cdot5050

=1{,}010{,}000.

Thus, the correct answer is E.