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The exercise asks to consider all natural numbers from 25 to 1915 (inclusive). Each number is multiplied by 3, and then 5 is added to each result. Finally, the sum of all these numbers is calculated. Let's break down each part of the task:

### (a) Represent the sum using summation notation:
Let \( n \) be the natural numbers from 25 to 1915. The transformation applied to each number is \( 3n + 5 \). The total sum can be represented as:

\[
\sum_{n=25}^{1915} (3n + 5)
\]

### (b) Deduce a general formula for the sum:
We need to find the formula for the sum of the sequence \( \sum_{n=25}^{1915} (3n + 5) \).

This summation can be split into two separate summations:

\[
\sum_{n=25}^{1915} (3n + 5) = 3 \sum_{n=25}^{1915} n + \sum_{n=25}^{1915} 5
\]

1. **For \( 3 \sum_{n=25}^{1915} n \):**
   The sum of the first \( N \) natural numbers is given by \( \frac{N(N+1)}{2} \). However, since the sum starts from 25, we can modify the formula:
   
   \[
   \sum_{n=25}^{1915} n = \sum_{n=1}^{1915} n - \sum_{n=1}^{24} n = \frac{1915(1916)}{2} - \frac{24(25)}{2}
   \]

2. **For \( \sum_{n=25}^{1915} 5 \):**
   The sum of a constant 5 added for every term from 25 to 1915 is simply \( 5 \times (1915 - 25 + 1) \).

### (c) Calculate the result using the formula:
Using the general formula deduced in part (b), we can compute the exact value of the sum.

Let's calculate these values step by step.The total sum of the sequence from 25 to 1915, where each number is multiplied by 3 and then 5 is added, is **5,512,265**.

### Summary:
1. **Summation notation**: \( \sum_{n=25}^{1915} (3n + 5) \)
2. **Formula breakdown**:
   - \( 3 \sum_{n=25}^{1915} n + 5 \times (1915 - 25 + 1) \)
3. **Result**: The total sum is **5,512,265**.

Would you like more details on any part of this calculation or the formula? 

Here are five related questions you might find helpful:
1. How do we derive the formula for summing an arithmetic sequence?
2. Can we calculate the sum for a different range of numbers?
3. How does the formula change if we multiply by a different number instead of 3?
4. What would happen if the numbers started from 1 instead of 25?
5. How would the formula change if instead of adding 5, we added another constant?

**Tip**: Breaking down summations into smaller parts can simplify complex summation problems and make calculations easier.

Se consideran todos los números naturales desde 25 a 1915 (ellos incluidos). A cada uno de ellos se lo multiplica por 3 y luego se le suma 5. Finalmente, se suman todos esos números.

Calculate the sum of all natural numbers from 25 to 1915, where each number is transformed by multiplying by 3 and adding 5. This problem involves arithmetic series and summation techniques, suitable for high school students.

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