Question

Compute each of these double sums

$\begin{array}{r}\sum _{\mathrm{i}=1}^2 \sum _{\mathrm{j}=1}^3 (\mathrm{i}+\mathrm{j})\\ \sum _{\mathrm{i}=0}^2 \sum _{\mathrm{j}=0}^3 (2\mathrm{i}+3\mathrm{j})\\ \sum _{\mathrm{i}=1}^3 \sum _{\mathrm{j}=0}^2 \mathrm{i}\\ \sum _{\mathrm{i}=0}^2 \sum _{\mathrm{j}=1}^3 \mathrm{ij}\end{array}$

Short answer

Thus, the double sums are given below:

1. 21
2. 78
3. 18
4. 18

Step-by-step solution

Step 1:

$\text{Given:}\sum _{\mathrm{i}=1}^2 \sum _{j=1}^3 (\mathrm{i}+\mathrm{j})$

Compute this double sum by first computing the inner sum followed by the outer sum.

$\begin{array}{r}\sum _{i=1}^2 \sum _{j=1}^3 (i+j)\\ \sum _{i=1}^2 (i+1+i+2+i+3)\\ \sum _{i=1}^2 (3i+6)\\ 3\sum _{i=1}^3 (i+2)\\ =3(1+2+2+2)\\ =21\end{array}$

Step 2:

Given: $\sum _{i=0}^2 \sum _{j=0}^3 (2i+3j)$

Compute this double sum by first computing the inner sum followed by the outer sum.

$\begin{array}{r}\sum _{i=0}^2 \sum _{j=0}^3 (2i+3j)\\ \sum _{i=0}^2 (2i+0+2i+3+2i+6+2i+9)\\ \sum _{i=0}^2 (8i+18)\\ =8\left(0\right)+18+8\left(1\right)+18+8\left(2\right)+18\\ =78\end{array}$

Step 3:

Given$\sum _{i+1^3\sum _{+}^2}^i$

Compute this double sum by first computing the inner sum followed by the outer sum.

$\begin{array}{r}\sum _{\mathrm{i}=1}^3 \sum _{\mathrm{j}=0}^2 \mathrm{ii}\\ \sum _{\mathrm{i}=1}^3 (\mathrm{i}+\mathrm{i}+\mathrm{i})\\ =\sum _{\mathrm{i}=1}^3 \left(3\mathrm{i}\right)\\ =3\sum _{\mathrm{i}=1}^3 \left(\mathrm{i}\right)\\ =3(1+2+3)\\ =18\end{array}$

Step 4:

Given:$\sum _{i=0}^2 \sum _{j=1}^3 ij$

Compute this double sum by first computing the inner sum followed by the outer sum.

$$\begin{gathered} \sum _{\mathrm{i}=0}^2 \sum _{\mathrm{j}=1}^3 \mathrm{jj} \\ \begin{array}{r}\sum _{\mathrm{i}=0}^2 (\mathrm{i}+2\mathrm{i}+3\mathrm{i})\\ =\sum _{\mathrm{i}=0}^2 \left(6\mathrm{i}\right)\\ =6\sum _{\mathrm{i}=0}^2 \left(\mathrm{i}\right)\\ =3(0+1+2)\\ =18\end{array} \end{gathered}$$