2018 AMC 12B Problem 9

Below is the professionally curated solution for Problem 9 of the 2018 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 12B solutions, or check the answer key.

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Concepts:summation arithmetic sequence

Difficulty rating: 1620

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

$100{,}100$

$500{,}500$

$505{,}000$

$1{,}001{,}000$

$1{,}010{,}000$

Solution:

Splitting the sum, $\sum_{i=1}^{100}\sum_{j=1}^{100}(i+j)=\sum_{i=1}^{100}\sum_{j=1}^{100}i+\sum_{i=1}^{100}\sum_{j=1}^{100}j=100\sum_{i=1}^{100}i+100\sum_{j=1}^{100}j.$

Since $\sum_{k=1}^{100}k=5050,$ this equals $100\cdot5050+100\cdot5050=1{,}010{,}000.$

Thus, the correct answer is E.

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