This is a double summation problem. The expression is:

$\sum_{j=2}^3 \left( \sum_{i=1}^3 (i + j) \right).$

Step 1: Expand the inner summation

The inner summation is: $\sum_{i=1}^3 (i + j).$

Expanding this for $i = 1, 2, 3$, we get: $(i + j) = (1 + j) + (2 + j) + (3 + j) = 1 + j + 2 + j + 3 + j = 6 + 3j.$

Thus, the inner summation is: $\sum_{i=1}^3 (i + j) = 6 + 3j.$

Step 2: Substitute the result into the outer summation

Now substitute $6 + 3j$ into the outer summation: $\sum_{j=2}^3 (6 + 3j).$

Expand this summation for $j = 2, 3$: $(6 + 3 \cdot 2) + (6 + 3 \cdot 3) = (6 + 6) + (6 + 9) = 12 + 15 = 27.$

Final Answer:

$\boxed{27}$

Let me know if you'd like a detailed breakdown or explanation! Here are some follow-up questions related to this problem:

1. How does changing the limits of summation (e.g., $j=1$ to $j=3$) affect the result?
2. What happens if the summand is $(i \cdot j)$ instead of $(i + j)$?
3. Can you rewrite the double summation as a single summation?
4. How would the computation differ for $i$ and $j$ in different ranges?
5. How does the summation change if the outer sum starts from $j=1$ instead of $j=2$?

Tip:

Always compute the inner summation first in double sums, as it simplifies the problem significantly before moving to the outer sum!