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The image presents a problem in Spanish that asks for the calculation of a sum involving terms from a geometric series. The expression shown is:

\[
1 + (-a) + (-a)^2 + \cdots + (-a)^n
\]

This is a finite geometric series with the first term \( a_1 = 1 \) and the common ratio \( r = -a \). The sum of the first \( n+1 \) terms of a geometric series is given by the formula:

\[
S_n = \frac{a_1(1 - r^{n+1})}{1 - r}
\]

Here, \( a_1 = 1 \) and \( r = -a \), so the sum formula becomes:

\[
S_n = \frac{1 - (-a)^{n+1}}{1 - (-a)}
\]

Simplifying further:

\[
S_n = \frac{1 - (-a)^{n+1}}{1 + a}
\]

This would be the sum of the series.

Would you like me to explain any further details about this sum or the derivation? Let me know if you'd like to solve a specific case of this series.

Here are five related questions:
1. How does the sum formula change for an infinite geometric series?
2. How do we determine whether a geometric series converges or diverges?
3. What happens to the sum if \( a = 1 \) in this specific series?
4. How does the sign of \( a \) affect the behavior of the series?
5. Can the same formula be applied if \( a \) is a complex number?

**Tip:** Always double-check the common ratio of the geometric series to ensure you're using the correct formula for finite or infinite sums.

Calcula la siguiente suma. (Supongamos que los términos son los términos consecutivos de una secuencia aritmética): 1 + (-a) + (-a)^2 + ... + (-a)^n

Learn how to calculate the sum of a geometric series 1 + (-a) + (-a)^2 + ... + (-a)^n. This problem involves key algebraic concepts such as geometric series and the sum formula, suitable for Grades 8-10.

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