geometric series: 

1. Find the sum of the first 20 terms geometric series 2, 4, 8, 16, 32, ...

2. Find the sum of the first 5 terms of the geometric series which, the first term is 2 and the common ratio is 3.

Let's solve both parts one by one.

### 1. Find the sum of the first 20 terms of the geometric series: 2, 4, 8, 16, 32, ...

This is a geometric series where:
- First term \( a = 2 \),
- Common ratio \( r = \frac{4}{2} = 2 \),
- Number of terms \( n = 20 \).

The formula to find the sum \( S_n \) of the first \( n \) terms of a geometric series is:

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

Substituting the values:

\[
S_{20} = 2 \frac{1 - 2^{20}}{1 - 2}
\]

\[
S_{20} = 2 \times \frac{1 - 1048576}{-1}
\]

\[
S_{20} = 2 \times 1048575 = 2097150
\]

So, the sum of the first 20 terms is:

\[
S_{20} = 2,097,150
\]

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### 2. Find the sum of the first 5 terms of the geometric series where the first term is 2 and the common ratio is 3.

In this case:
- First term \( a = 2 \),
- Common ratio \( r = 3 \),
- Number of terms \( n = 5 \).

Using the same formula for the sum of a geometric series:

\[
S_5 = a \frac{1 - r^n}{1 - r}
\]

Substitute the values:

\[
S_5 = 2 \frac{1 - 3^5}{1 - 3}
\]

\[
S_5 = 2 \times \frac{1 - 243}{-2}
\]

\[
S_5 = 2 \times \frac{-242}{-2}
\]

\[
S_5 = 2 \times 121 = 242
\]

So, the sum of the first 5 terms is:

\[
S_5 = 242
\]

---

Would you like more details on how the formula works or have any questions?

Here are 5 related questions to explore further:

1. What is the general form of a geometric series?
2. How would you find the sum of an infinite geometric series?
3. What happens to the sum of a geometric series if the common ratio is negative?
4. How does the sum change if the common ratio is a fraction (less than 1)?
5. How can we derive the sum formula for a geometric series?

**Tip:** The sum formula for a geometric series only works for \( r \neq 1 \). If \( r = 1 \), the series is arithmetic.

1. Find the sum of the first 20 terms of the geometric series: 2, 4, 8, 16, 32, ...
2. Find the sum of the first 5 terms of the geometric series where the first term is 2 and the common ratio is 3.

Discover how to calculate the sum of the first 20 terms of a geometric series starting with 2, 4, 8, 16, 32, and the sum of the first 5 terms of a geometric series where the first term is 2 and the common ratio is 3. Step-by-step solutions provided, suitable for Grades 9-12.

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