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The image presents an arithmetic sequence where the terms follow a specific pattern. The sum requested involves the following sequence:

\[ 2 + 7 + 12 + \dots + 522 \]

This is an arithmetic sequence where:
- The first term \(a = 2\),
- The common difference \(d = 7 - 2 = 5\),
- The last term \(l = 522\).

To find the sum of an arithmetic sequence, we use the formula:

\[
S_n = \frac{n}{2} \times (a + l)
\]

Where:
- \(S_n\) is the sum,
- \(n\) is the number of terms,
- \(a\) is the first term,
- \(l\) is the last term.

First, we need to find \(n\), the number of terms in the sequence, using the formula for the \(n\)-th term of an arithmetic sequence:

\[
l = a + (n-1) \times d
\]

Solving for \(n\):

\[
522 = 2 + (n-1) \times 5
\]

Let me calculate that for you.The number of terms in the sequence \( n = 105 \).

Now, we can calculate the sum of the sequence using the formula:

\[
S_n = \frac{n}{2} \times (a + l)
\]

Substituting the known values:

\[
S_{105} = \frac{105}{2} \times (2 + 522)
\]

Let me calculate the sum for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the sum manually using the formula:

\[
S_{105} = \frac{105}{2} \times (2 + 522) = \frac{105}{2} \times 524 = 105 \times 262 = 27,510
\]

So, the sum of the arithmetic sequence is \( 27,510 \).

Would you like any further explanation or have additional questions?

Here are five related questions:
1. How do you derive the formula for the sum of an arithmetic series?
2. What is the difference between arithmetic and geometric sequences?
3. Can you calculate the sum if the sequence starts with a different first term?
4. How would you find the sum of an infinite arithmetic sequence?
5. How can the sum formula be applied to other real-world problems?

**Tip:** Always check if the sequence you're working with is arithmetic by confirming the common difference between terms.

Calculate the following sum. (Assume that the terms are consecutive terms of an arithmetic sequence.)

2 + 7 + 12 + ... + 522 = ?

Learn how to calculate the sum of an arithmetic sequence from 2 to 522. This guide covers the formulas for finding the nth term and the sum of an arithmetic series, suitable for high school students.

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