Evaluate the sum of the first 20 terms of the series:
 3 + 5 + 9 + 17 + 33 + 65 + ...

To evaluate the sum of the first 20 terms of the series \( 3 + 5 + 9 + 17 + 33 + 65 + \dots \), let's first analyze the pattern in the series.

### Step 1: Identify the pattern
The given series is:
\[
3, 5, 9, 17, 33, 65, \dots
\]

Let's observe the difference between consecutive terms:
\[
5 - 3 = 2 \\
9 - 5 = 4 \\
17 - 9 = 8 \\
33 - 17 = 16 \\
65 - 33 = 32
\]

The differences between the terms appear to follow a pattern:
\[
2, 4, 8, 16, 32, \dots
\]
This is a geometric progression with a common ratio of 2.

Thus, the series is not an arithmetic or geometric progression but appears to be based on exponential differences.

### Step 2: Express the general term
Let's denote the \(n\)-th term of the series by \( a_n \). We can express \( a_n \) as:
\[
a_n = 3 + \sum_{k=1}^{n-1} 2^k
\]

The sum of a geometric series is given by:
\[
\sum_{k=1}^{n-1} 2^k = 2^1 + 2^2 + \dots + 2^{n-1} = 2(2^{n-1} - 1)
\]

Thus:
\[
a_n = 3 + 2(2^{n-1} - 1)
\]

Simplifying further:
\[
a_n = 2^n + 1
\]

### Step 3: Sum the first 20 terms
Now that we have the formula for the \(n\)-th term \( a_n = 2^n + 1 \), the sum \( S_{20} \) of the first 20 terms is given by:
\[
S_{20} = \sum_{n=1}^{20} (2^n + 1)
\]
This can be split into two separate sums:
\[
S_{20} = \sum_{n=1}^{20} 2^n + \sum_{n=1}^{20} 1
\]
The first sum is the sum of a geometric series:
\[
\sum_{n=1}^{20} 2^n = 2(2^{20} - 1)
\]
The second sum is simply the sum of 20 ones:
\[
\sum_{n=1}^{20} 1 = 20
\]
Thus:
\[
S_{20} = 2(2^{20} - 1) + 20
\]
Calculating the value:
\[
2^{20} = 1,048,576
\]
So:
\[
S_{20} = 2(1,048,576 - 1) + 20 = 2(1,048,575) + 20 = 2,097,150 + 20 = 2,097,170
\]

### Final Answer:
The sum of the first 20 terms of the series is \( \boxed{2,097,170} \).

Would you like more details or have any questions? 

Here are some related problems to explore:

1. Find the sum of the first 15 terms of the same series.
2. Determine the 25th term of the series.
3. Derive the sum of the first 30 terms of a similar series with different initial values.
4. Investigate the sum of the first 20 terms if each difference was multiplied by 3 instead of 2.
5. Explore what happens if you start the series with a different initial value.

**Tip:** Identifying the pattern or formula for the general term of a series is often the key step in solving summation problems like these.

Learn how to evaluate the sum of the first 20 terms of the series 3 + 5 + 9 + 17 + 33 + 65 with a detailed step-by-step solution. This problem covers series analysis, geometric progression, and summation formulas. Suitable for high school students.

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