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Let's solve the geometric series problems one by one.

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

The sum of a finite geometric series is given by:
\[
S_n = a \frac{1 - r^n}{1 - r}
\]
Substitute the values:
\[
S_9 = 1 \cdot \frac{1 - (-4)^9}{1 - (-4)} = \frac{1 - (-262144)}{5} = \frac{262145}{5} = 52429
\]
So, the sum is **52,429**.

---

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

Using the formula for the sum:
\[
S_8 = 2 \cdot \frac{1 - (-5)^8}{1 - (-5)} = 2 \cdot \frac{1 - 390625}{6} = 2 \cdot \frac{-390624}{6} = 2 \cdot (-65104) = -130208
\]
The sum is **-130,208**.

---

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

Using the sum formula:
\[
S_6 = 1 \cdot \frac{1 - 2^6}{1 - 2} = \frac{1 - 64}{-1} = \frac{-63}{-1} = 63
\]
The sum is **63**.

---

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

Using the formula:
\[
S_7 = 1 \cdot \frac{1 - (-5)^7}{1 - (-5)} = \frac{1 - (-78125)}{6} = \frac{78126}{6} = 13021
\]
The sum is **13,021**.

---

### 5. \( \sum_{k=1}^{7} 4^{k-1} \)
This is a geometric series where:
- First term \( a = 4^{0} = 1 \)
- Common ratio \( r = 4 \)
- Number of terms \( n = 7 \)

Using the sum formula:
\[
S_7 = 1 \cdot \frac{1 - 4^7}{1 - 4} = \frac{1 - 16384}{-3} = \frac{-16383}{-3} = 5461
\]
The sum is **5,461**.

---

Would you like details on any of these solutions or more explanation? Let me know!

Here are some additional questions to explore:
1. How is the common ratio of a geometric series determined?
2. What happens to the sum of an infinite geometric series if \( |r| < 1 \)?
3. Can the sum formula for geometric series be derived? How?
4. What real-world applications can you think of for geometric series?
5. How would you calculate the sum if you had \( n \) for a very large value?

**Tip**: Always check the sign and magnitude of the common ratio carefully, as it affects both the sum and the series behavior!

Evaluate each geometric series described, including 1 - 4 + 16 - 64..., n = 9 and others.

Learn how to evaluate geometric series problems like 1 - 4 + 16 - 64..., with n = 9, using the formula for finite sums. This guide covers geometric progression concepts, common ratio calculations, and step-by-step solutions for students in Grades 10-12.

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