The image contains a geometric series problem where the first term ($a$), the common ratio ($r$), and the number of terms ($n$) are given. You are asked to determine the sum of the first $n$ terms of the series.

The formula for the sum of the first $n$ terms ($S_n$) of a geometric series is:

$S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{if} \quad r \neq 1$

Here is how you can solve the problem for each case:

a. $a = 4$, $r = 2$, $n = 10$

$S_n = \frac{4(1 - 2^{10})}{1 - 2} = \frac{4(1 - 1024)}{-1} = \frac{4(-1023)}{-1} = 4092$

b. $a = 16$, $r = 1.5$, $n = 18$

$S_n = \frac{16(1 - (1.5)^{18})}{1 - 1.5}$ This will need to be calculated.

c. $a = -81$, $r = 0.9$, $n = 20$

$S_n = \frac{-81(1 - (0.9)^{20})}{1 - 0.9}$ This also needs to be calculated.

d. $a = 1.25$, $r = 3$, $n = 22$

$S_n = \frac{1.25(1 - 3^{22})}{1 - 3}$ This will also need calculation.

Let me calculate the exact values for the rest of the cases.Here are the sums of the first $n$ terms for each of the geometric series:

1. a = 4, r = 2, n = 10: $S_n = 4092$
2. a = 16, r = 1.5, n = 18: $S_n \approx 47260.54$
3. a = -81, r = 0.9, n = 20: $S_n \approx -711.52$
4. a = 1.25, r = 3, n = 22: $S_n \approx 19,613,162,255$

If you need more details or have any questions, feel free to ask!

Related questions:

1. What happens when the ratio $r = 1$?
2. How does the sign of $a$ affect the geometric sum?
3. What is the behavior of the geometric sum when $r$ is negative?
4. How do you derive the formula for the sum of a geometric series?
5. What is the difference between finite and infinite geometric series?

Tip:

In a geometric series, if the absolute value of $r$ is less than 1, the series converges as $n$ increases.