To solve the summation $\sum_{i=1}^{14} (9i + 3)$, we can follow these steps:

1. Separate the summation: $\sum_{i=1}^{14} (9i + 3) = \sum_{i=1}^{14} 9i + \sum_{i=1}^{14} 3$

2. Simplify each part:

   • For the first part, $\sum_{i=1}^{14} 9i$: $9 \sum_{i=1}^{14} i$ The summation of the first $n$ natural numbers is given by the formula: $\sum_{i=1}^n i = \frac{n(n+1)}{2}$ So, $\sum_{i=1}^{14} i = \frac{14 \cdot 15}{2} = 105$ Therefore, $9 \sum_{i=1}^{14} i = 9 \cdot 105 = 945$

   • For the second part, $\sum_{i=1}^{14} 3$: $3 \sum_{i=1}^{14} 1 = 3 \cdot 14 = 42$

3. Add the two parts together: $945 + 42 = 987$

Therefore, the value of the summation $\sum_{i=1}^{14} (9i + 3)$ is $\boxed{987}$.

Would you like more details or have any questions?

Here are 8 related questions that expand on this topic:

1. How do you derive the formula for the sum of the first $n$ natural numbers?
2. Can you generalize the process for finding $\sum_{i=1}^{n} (ai + b)$?
3. How does the linearity of summation help simplify complex summations?
4. What is the formula for the sum of the first $n$ squares and how is it derived?
5. How can you extend these techniques to summations involving quadratic expressions?
6. What are some real-world applications of arithmetic summations?
7. How would you compute the sum of an arithmetic series with a different starting index?
8. What are geometric series and how do their summation formulas differ from arithmetic series?

Tip: Always break down complex summations into simpler parts to make calculations easier.