Multiple Summations

$\begin{aligned} S & = \sum_{i=0}^2 \sum_{j=0}^{\color{#D61F06}3} (2{\color{#3D99F6}i}+3j) & \small \color{#3D99F6} \text{Note that } i \text{ is independent of } j \text{ and is treated as a constant.} \\ & = \sum_{i=0}^{\color{#3D99F6}2} \left(2 {\color{#3D99F6}i} ({\color{#D61F06}3}+1)+\frac{3({\color{#D61F06}3})({\color{#D61F06}3}+1)}2 \right) \\ & = \sum_{i=0}^{\color{#3D99F6}2} \left(8 {\color{#3D99F6}i} + 18 \right) \\ & = \frac{8 ({\color{#3D99F6}2})({\color{#3D99F6}2}+1)}2 + 18({\color{#3D99F6}2}+1) \\ & = 24 + 54 = \boxed{78} \end{aligned}$

$\text{Precisely,}$

$\large \mathfrak{S} = \sum_{i=0}^{2} \sum_{j=0}^{3} (2i+3j)$

$\large \mathfrak{S} = \sum_{i=0}^{2} (2i + 2i + 3 + 2i + 6 + 2i + 9)$

$\large \mathfrak{S} = \sum_{i=0}^{2} (8i+18)$

$\large \mathfrak{S} = 8.3 + 18.3 = \boxed{78}$