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The question asks about an arithmetic progression (AP) and gives the following information:

- The sum of the first 7 terms (\(S_7\)) is 49.
- The sum of the first 17 terms (\(S_{17}\)) is 289.

We are required to find the sum of the first 20 terms (\(S_{20}\)).

### Formula for the sum of terms in an AP:
The sum of the first \(n\) terms of an AP is given by:
\[
S_n = \frac{n}{2} \times (2a + (n-1)d)
\]
where:
- \(S_n\) is the sum of the first \(n\) terms,
- \(a\) is the first term,
- \(d\) is the common difference,
- \(n\) is the number of terms.

### Step 1: Use the sum formula for \(S_7\) and \(S_{17}\)

From the first equation:
\[
S_7 = \frac{7}{2} \times (2a + 6d) = 49
\]
This simplifies to:
\[
7(2a + 6d) = 98 \implies 2a + 6d = 14 \quad \text{(Equation 1)}
\]

For the second equation:
\[
S_{17} = \frac{17}{2} \times (2a + 16d) = 289
\]
This simplifies to:
\[
17(2a + 16d) = 578 \implies 2a + 16d = 34 \quad \text{(Equation 2)}
\]

### Step 2: Solve for \(a\) and \(d\)

Subtract Equation 1 from Equation 2:
\[
(2a + 16d) - (2a + 6d) = 34 - 14
\]
\[
10d = 20 \implies d = 2
\]

Substitute \(d = 2\) into Equation 1:
\[
2a + 6(2) = 14 \implies 2a + 12 = 14 \implies 2a = 2 \implies a = 1
\]

### Step 3: Find the sum of the first 20 terms

Now that we know \(a = 1\) and \(d = 2\), we can use the sum formula to find \(S_{20}\):
\[
S_{20} = \frac{20}{2} \times (2(1) + (20-1)(2)) = 10 \times (2 + 38) = 10 \times 40 = 400
\]

### Final Answer:
The sum of the first 20 terms is **400**, so the correct option is **A**.

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Would you like a deeper explanation on any part? Here are some follow-up questions you might find helpful:

1. How can I derive the formula for the sum of an arithmetic progression?
2. Why do we subtract the two equations to find the common difference \(d\)?
3. How would the problem change if the common difference \(d\) was negative?
4. What are the general characteristics of an arithmetic progression?
5. How would you solve for the \(n\)-th term directly in this AP?

**Tip:** Always double-check the order of terms in your formulas to avoid simple algebraic mistakes!

If the sum of the first 7 terms of an AP is 49 and the first 17 terms is 289, find the sum of its first 20 terms.

Learn how to find the sum of the first 20 terms of an arithmetic progression (AP) using the sum formula. The problem provides the sum of the first 7 and 17 terms. Step-by-step solution for high school students.

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