https://file.aiquickdraw.com/mathbot/file/1727347282031f6lzov88.jpg

Let's solve the geometric series problems provided in the image step-by-step:

### 1. \( 1 - 4 + 16 - 64 + \ldots \), \( n = 9 \)
This is a geometric series where:
- First term \( a = 1 \)
- Common ratio \( r = -4 \)
- Number of terms \( n = 9 \)

The sum of a finite geometric series is given by:
\[
S_n = \frac{a(1 - r^n)}{1 - r}
\]

Substituting the values:
\[
S_9 = \frac{1(1 - (-4)^9)}{1 - (-4)} = \frac{1(1 + 262144)}{5} = \frac{262145}{5} = 52429
\]

So, the sum is **52429**.

### 2. \( 2 - 10 + 50 - 250 + \ldots \), \( n = 8 \)
This is also a geometric series where:
- First term \( a = 2 \)
- Common ratio \( r = -5 \)
- Number of terms \( n = 8 \)

Applying the formula for the sum:
\[
S_n = \frac{a(1 - r^n)}{1 - r}
\]
Substituting the values:
\[
S_8 = \frac{2(1 - (-5)^8)}{1 - (-5)} = \frac{2(1 - 390625)}{6} = \frac{2(-390624)}{6} = \frac{-781248}{6} = -130208
\]

So, the sum is **-130208**.

### 3. \( 1 + 2 + 4 + 8 + \ldots \), \( n = 6 \)
This is a geometric series where:
- First term \( a = 1 \)
- Common ratio \( r = 2 \)
- Number of terms \( n = 6 \)

Applying the formula for the sum:
\[
S_n = \frac{a(1 - r^n)}{1 - r}
\]
Substituting the values:
\[
S_6 = \frac{1(1 - 2^6)}{1 - 2} = \frac{1(1 - 64)}{-1} = \frac{-63}{-1} = 63
\]

So, the sum is **63**.

### 4. \( 1 - 5 + 25 - 125 + \ldots \), \( n = 7 \)
This is a geometric series where:
- First term \( a = 1 \)
- Common ratio \( r = -5 \)
- Number of terms \( n = 7 \)

Using the sum formula:
\[
S_7 = \frac{1(1 - (-5)^7)}{1 - (-5)} = \frac{1(1 + 78125)}{6} = \frac{78126}{6} = 13021
\]

So, the sum is **13021**.

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Would you like me to continue solving the rest of the problems from 5 to 10? Let me know if you need further details on the calculations! 

Here are some related questions to consider:
1. How do you determine the common ratio of a geometric series?
2. What happens when the common ratio is negative in a geometric series?
3. How does the sum formula change for infinite geometric series?
4. Can a geometric series have a common ratio of 1?
5. How would you apply this method to a geometric series with fractions?

**Tip**: Always double-check the number of terms \( n \) when applying the geometric series sum formula to avoid calculation errors.

Evaluate each geometric series described in the provided image (e.g., 1 - 4 + 16 - 64, n = 9; 2 - 10 + 50 - 250, n = 8).

Learn how to evaluate finite geometric series step-by-step, including examples like 1 - 4 + 16 - 64 with n = 9 and 2 - 10 + 50 - 250 with n = 8. This covers the formula for summing geometric progressions and is suitable for high school students.

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