The sum of the first 10 terms of an arithmetic sequence is 175. If the first term is 3, find the common difference.

To find the common difference of an arithmetic sequence given the sum of the first 10 terms and the first term, we can use the formula for the sum of an arithmetic sequence:

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

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.

Given:
\[ S_{10} = 175 \]
\[ a = 3 \]
\[ n = 10 \]

Plugging these values into the formula:

\[ 175 = \frac{10}{2} (2 \cdot 3 + (10-1)d) \]

Simplify the equation:

\[ 175 = 5 (6 + 9d) \]

Divide both sides by 5:

\[ 35 = 6 + 9d \]

Subtract 6 from both sides:

\[ 29 = 9d \]

Solve for \( d \):

\[ d = \frac{29}{9} \]

Therefore, the common difference is:

\[ d = \frac{29}{9} \]

If you have any questions or want more details, please let me know.

Here are some related questions you might be interested in:

1. How do you find the sum of the first \( n \) terms of an arithmetic sequence?
2. What is the formula for the \( n \)-th term of an arithmetic sequence?
3. How does changing the common difference affect an arithmetic sequence?
4. Can the common difference of an arithmetic sequence be negative? What does it mean?
5. How do you determine the number of terms in an arithmetic sequence given the first term, common difference, and the last term?
6. How do you find the common difference if the first term and the \( n \)-th term are given?
7. What are some real-life applications of arithmetic sequences?
8. How do you derive the formula for the sum of an arithmetic sequence?

**Tip:** Always double-check your calculations to ensure accuracy, especially when dealing with multiple steps in algebraic manipulation.

Learn how to find the common difference of an arithmetic sequence when given the sum of the first 10 terms and the first term. Step-by-step solution using the formula for the sum of an arithmetic sequence. Suitable for high school students.

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