the third term of an arithmetic sequence is -24 and the seventh term is -40. What is the sum of the first 10 term?

To solve this problem, we need to first determine the first term \( a_1 \) and the common difference \( d \) of the arithmetic sequence.

### Step 1: Set up the equations
Given:
- The third term \( a_3 = -24 \)
- The seventh term \( a_7 = -40 \)

The general formula for the \( n \)th term of an arithmetic sequence is:
\[
a_n = a_1 + (n - 1) \cdot d
\]

Using this formula:
1. For the third term (\( n = 3 \)):
\[
a_3 = a_1 + 2d = -24 \quad \text{(Equation 1)}
\]

2. For the seventh term (\( n = 7 \)):
\[
a_7 = a_1 + 6d = -40 \quad \text{(Equation 2)}
\]

### Step 2: Solve the system of equations
Subtract Equation 1 from Equation 2:
\[
(a_1 + 6d) - (a_1 + 2d) = -40 - (-24)
\]
\[
4d = -16 \implies d = -4
\]

Substitute \( d = -4 \) back into Equation 1:
\[
a_1 + 2(-4) = -24
\]
\[
a_1 - 8 = -24 \implies a_1 = -16
\]

### Step 3: Find the sum of the first 10 terms
The sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is given by:
\[
S_n = \frac{n}{2} \cdot (2a_1 + (n - 1) \cdot d)
\]
Here, \( n = 10 \), \( a_1 = -16 \), and \( d = -4 \).

Substitute the values:
\[
S_{10} = \frac{10}{2} \cdot (2(-16) + 9(-4))
\]
\[
S_{10} = 5 \cdot (-32 - 36)
\]
\[
S_{10} = 5 \cdot (-68) = -340
\]

### Final Answer:
The sum of the first 10 terms is **-340**.

Would you like more details on any part of this solution, or do you have any questions? Here are some related questions:

1. How would the sum change if the sequence had a positive common difference?
2. How do you determine the sum if the number of terms is not given directly?
3. What is the significance of the common difference in an arithmetic sequence?
4. How can you find the middle term of an arithmetic sequence?
5. What are some real-world applications of arithmetic sequences?

**Tip:** In arithmetic sequences, knowing any two terms allows you to find the first term and the common difference, which are key to determining other properties of the sequence.

Learn how to find the sum of the first 10 terms in an arithmetic sequence with a detailed step-by-step solution. This problem covers key concepts such as arithmetic sequences, nth term formula, and the sum formula for arithmetic sequences. Suitable for high school students.

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