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The problem asks to calculate the sum of the sequence \( 1 + (-4) + (-9) + \dots + (-474) \), which is part of an arithmetic sequence.

### Step 1: Identify the arithmetic sequence components
In an arithmetic sequence, the general form of the \(n\)-th term is given by:
\[
a_n = a_1 + (n-1) \cdot d
\]
where:
- \(a_1 = 1\) is the first term,
- \(d = -5\) is the common difference (since \( (-4) - 1 = -5\), and \( (-9) - (-4) = -5\)).

The last term of the sequence is \(a_n = -474\).

### Step 2: Find the number of terms
To find the number of terms \(n\), we use the formula for the \(n\)-th term:
\[
a_n = a_1 + (n-1) \cdot d
\]
Substitute the values:
\[
-474 = 1 + (n-1) \cdot (-5)
\]
Simplifying:
\[
-474 = 1 - 5(n-1)
\]
\[
-474 = 1 - 5n + 5
\]
\[
-474 = 6 - 5n
\]
\[
-480 = -5n
\]
\[
n = \frac{480}{5} = 96
\]
So, there are 96 terms in the sequence.

### Step 3: Use the sum formula for an arithmetic sequence
The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by:
\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\]
Substituting the known values:
\[
S_{96} = \frac{96}{2} \cdot (1 + (-474))
\]
\[
S_{96} = 48 \cdot (-473)
\]
\[
S_{96} = -22704
\]

### Final Answer:
The sum of the sequence is \(-22,704\).

Let me know if you would like more details or have any questions! 

Here are 5 related questions:
1. What is the general formula for the sum of an arithmetic sequence?
2. How does the common difference affect the sum of an arithmetic series?
3. Can an arithmetic sequence have a positive sum with a negative common difference?
4. How do we determine the number of terms in a sequence if only the first and last terms are given?
5. What happens if the common difference in an arithmetic sequence is positive?

**Tip:** The sum of an arithmetic sequence can be calculated faster by recognizing symmetry when the sequence has both positive and negative terms!

Calcular la suma 1 + (-4) + (-9) + ... + (-474), asumiendo que los términos son consecutivos de una sucesión aritmética.

Learn how to calculate the sum of an arithmetic sequence starting at 1 and ending at -474, with a common difference of -5. The problem involves applying key formulas for arithmetic sequences and determining the number of terms in the sequence.

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