This is a summation problem, where you evaluate the sum of the expression $i^3 - 2 + 3i$ for $i$ ranging from 3 to 5.

The formula for this summation is: $\sum_{i=3}^{5} \left( i^3 - 2 + 3i \right) = (3^3 - 2 + 3 \cdot 3) + (4^3 - 2 + 3 \cdot 4) + (5^3 - 2 + 3 \cdot 5)$

Step-by-Step Solution:

1. Evaluate each term individually:

   • For $i = 3$: $3^3 - 2 + 3 \cdot 3 = 27 - 2 + 9 = 34$
   • For $i = 4$: $4^3 - 2 + 3 \cdot 4 = 64 - 2 + 12 = 74$
   • For $i = 5$: $5^3 - 2 + 3 \cdot 5 = 125 - 2 + 15 = 138$

2. Add the results: $34 + 74 + 138 = 246$

Final Answer:

$\boxed{246}$

Would you like further clarification or a different approach to this problem?

Here are 5 related questions to explore:

1. How would the solution change if the summation limits were different (e.g., $i = 2$ to $i = 5$)?
2. Can you write a general formula for summations of cubic terms $\sum_{i=a}^b i^3$?
3. How would this problem change if the summand had a quadratic term instead of a cubic one?
4. Can you explain how summation properties (e.g., linearity) can simplify such problems?
5. What are the practical applications of evaluating summations in real-world problems?

Tip: Always break down summations into individual terms to evaluate them efficiently, especially for small ranges of $i$.