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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.

Learn how to solve the summation problem \(\sum_{i=1}^{14} (9i + 3)\) with a detailed step-by-step solution. This problem covers arithmetic series and the formula for the sum of a linear expression. Suitable for students in Grades 10-12.

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