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!