To find the sum of the series:

$S = -1 - 5 - 25 - 125 - \dots - 5^{n-1},$

we recognize this as a geometric series with:

• First term ($a$) = $-1$,
• Common ratio ($r$) = $5$,
• Number of terms ($n$).

Step 1: Formula for the sum of a geometric series

The sum of a finite geometric series is given by:

$S_n = a \frac{1 - r^n}{1 - r}, \quad \text{if } r \neq 1.$

Here:

• $a = -1$,
• $r = 5$,
• $n$ is the number of terms.

Substitute the values:

$S_n = (-1) \frac{1 - 5^n}{1 - 5}.$

Step 2: Simplify the denominator

$1 - 5 = -4.$

Thus, the sum becomes:

$S_n = (-1) \frac{1 - 5^n}{-4}.$

Step 3: Simplify further

$S_n = \frac{1 - 5^n}{4}.$

Final Answer:

The sum of the series is:

$S_n = \frac{1 - 5^n}{4}.$

If you have specific $n$ in mind, let me know to compute the numerical sum! Would you like further explanation?

---

Expanding Questions:

1. What is the sum when $n = 5$ or $n = 10$?
2. How does the sum change as $n \to \infty$?
3. Can you derive this result using summation notation directly?
4. What happens if the ratio $r$ is a fraction instead of 5?
5. How would the formula differ if the series started with $-5$ instead of $-1$?