This is a geometric-like series with a common ratio pattern where each term increases by a factor that doubles. To solve this, we first analyze the pattern and determine the general term of the sequence, then find the sum of the first 20 terms.

Step 1: Observe the Series

The sequence is:

$3, 5, 9, 17, 33, 65, \dots$

By observing the differences between consecutive terms:

$5 - 3 = 2,\quad 9 - 5 = 4,\quad 17 - 9 = 8,\quad 33 - 17 = 16,\quad 65 - 33 = 32$

The difference doubles at each step, suggesting a geometric progression in the differences. Hence, the sequence itself is a second-order recurrence relation.

Step 2: General Formula of the Series

Let the $n$-th term of the sequence be $a_n$. The pattern indicates:

$a_{n+1} = a_n + 2^{n-1}, \quad \text{where } a_1 = 3.$

To find $a_n$, we sum the differences:

$a_n = 3 + \sum_{k=1}^{n-1} 2^k$

Step 3: Simplify Using the Sum of a Geometric Series

The sum of the first $m$ terms of $2^k$ is given by:

$\sum_{k=1}^m 2^k = 2(2^m - 1)$

For $a_n$, this becomes:

$a_n = 3 + (2 \cdot (2^{n-1} - 1)) = 3 + 2 \cdot 2^{n-1} - 2 = 2^{n} + 1$

Step 4: Find the Sum of the First 20 Terms

The sum of the first $N$ terms is:

$S_N = \sum_{n=1}^N a_n = \sum_{n=1}^N (2^n + 1)$

Split this into two sums:

$S_N = \sum_{n=1}^N 2^n + \sum_{n=1}^N 1$

• The sum of $2^n$ for $n = 1$ to $N$ is:

$\sum_{n=1}^N 2^n = 2(2^N - 1)$

• The sum of $1$ for $n = 1$ to $N$ is:

$\sum_{n=1}^N 1 = N$

Combine these:

$S_N = 2(2^N - 1) + N$

For $N = 20$:

$S_{20} = 2(2^{20} - 1) + 20$

Step 5: Calculate the Result

$2^{20} = 1,048,576$

Substitute:

$S_{20} = 2(1,048,576 - 1) + 20 = 2(1,048,575) + 20 = 2,097,150 + 20 = 2,097,170$

Final Answer:

The sum of the first 20 terms is:

$\boxed{2,097,170}$

Would you like a detailed explanation of any step?

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Questions to Expand:

1. How can you derive a recurrence relation for a different sequence?
2. What are the properties of a geometric progression in terms of summation?
3. How is the formula for the sum of powers of 2 derived?
4. Can this series be extended to a general $k$-step progression?
5. How does this sequence relate to exponential growth in real-life contexts?

Tip:

When analyzing sequences, always check for consistent differences or ratios between terms—they hint at arithmetic or geometric patterns.