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.