The problem involves finding the sum of a geometric series:

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

Step 1: Identify the geometric series

The series can be rewritten as: $S = -(1 + 5 + 5^2 + 5^3 + \dots + 5^{n-1})$

This is a geometric series where:

• The first term ($a$) is $1$,
• The common ratio ($r$) is $5$,
• The number of terms is $n$.

Step 2: Formula for the sum of a geometric series

The sum of a geometric series is given by: $\text{Sum} = a \frac{r^n - 1}{r - 1}$

Substitute $a = 1$, $r = 5$, and the number of terms $n$: $\text{Sum of positive terms} = \frac{5^n - 1}{5 - 1} = \frac{5^n - 1}{4}$

Step 3: Include the negative sign

Since the series is multiplied by $-1$, the total sum becomes: $S = -\frac{5^n - 1}{4}$

Final Answer:

$\boxed{S = -\frac{5^n - 1}{4}}$

Let me know if you'd like a step-by-step example with specific $n$ or have any further questions!

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Questions to Expand Understanding:

1. What happens if the common ratio is changed to another number?
2. How does the formula for the sum of an infinite geometric series differ from this?
3. Can we derive the formula for a geometric series from first principles?
4. How does the sum behave as $n$ becomes very large?
5. Can we rewrite this series in terms of a recursive formula?

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Tip:

When dealing with series, always check whether it's finite or infinite, as the formulas differ significantly!