The sum of the first 40 terms of an arithmetic sequence is 3220. The first term is 2.5. Find the common difference.

We can solve this problem using the formula for the sum of an arithmetic sequence:

\[
S_n = \frac{n}{2} \cdot (2a + (n - 1)d)
\]

Where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference.

Given:
- \( S_{40} = 3220 \) (sum of the first 40 terms),
- \( a = 2.5 \) (the first term),
- \( n = 40 \) (number of terms),
- \( d \) is unknown.

Substituting the given values into the formula:

\[
3220 = \frac{40}{2} \cdot (2(2.5) + (40 - 1)d)
\]

Simplify:

\[
3220 = 20 \cdot (5 + 39d)
\]

\[
3220 = 20 \cdot (5 + 39d)
\]

\[
3220 = 100 + 780d
\]

Now, subtract 100 from both sides:

\[
3120 = 780d
\]

Solve for \( d \):

\[
d = \frac{3120}{780} = 4
\]

Thus, the common difference \( d \) is 4.

### Do you need any more details or have questions?

Here are 5 related questions:
1. How do we find the \( n \)-th term of an arithmetic sequence?
2. What is the 40th term of this sequence with the common difference of 4?
3. Can you derive the formula for the sum of an arithmetic sequence?
4. What is the sum of the first 50 terms of this sequence?
5. How does the common difference affect the behavior of an arithmetic sequence?

**Tip:** The common difference in an arithmetic sequence determines the rate at which each term increases or decreases!

Learn how to find the common difference of an arithmetic sequence where the sum of the first 40 terms is 3220 and the first term is 2.5. This step-by-step solution involves using the sum of arithmetic sequence formula. Ideal for Grades 9-10 students.

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