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.

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.