smc

Recall that the sum of the first $100$ positive integers is ${\sum }_{k=1}^{100}k=\frac{101\cdot 100}{2}=5050.$ It follows that $$\begin{array}{rl}\sum _{i=1}^{100}\sum _{j=1}^{100}(i+j)& =\sum _{i=1}^{100}\sum _{j=1}^{100}i+\sum _{i=1}^{100}\sum _{j=1}^{100}j\\ & =\sum _{i=1}^{100}(100i)+100\sum _{j=1}^{100}j\\ & =100\sum _{i=1}^{100}i+100\sum _{j=1}^{100}j\\ & =100\cdot 5050+100\cdot 5050\\ & =\boxed{1,010,000}.\end{array}$$