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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.

Diketahui suku pertama (a), rasio (r), dan banyaknya suku (n) dari deret geometri berikut ini. Tentukanlah jumlah n suku pertama dan deret!

Learn how to find the sum of the first n terms of a geometric series. This problem covers key concepts in geometric progressions, with examples using different first terms (a), common ratios (r), and number of terms (n). Suitable for high school students.

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