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:

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.