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:

$S_5 = 242$

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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.