The sum of the first and fifth terms of an arithmetic sequence is 40. The sum of the second and sixth terms is 52. Find the sum of the first ten terms.$$  $$

To find the sum of the first ten terms of an arithmetic sequence given the conditions, we will follow these steps:

1. **Identify the common terms and notations:**

   Let $ a $ be the first term and $ d $ be the common difference of the arithmetic sequence.

2. **Translate the given conditions into equations:**

   - The sum of the first and fifth terms is 40:
     \[
     a + (a + 4d) = 40
     \]
     Simplify this to:
     \[
     2a + 4d = 40 \quad \text{(Equation 1)}
     \]

   - The sum of the second and sixth terms is 52:
     \[
     (a + d) + (a + 5d) = 52
     \]
     Simplify this to:
     \[
     2a + 6d = 52 \quad \text{(Equation 2)}
     \]

3. **Solve the system of equations:**

   From Equation 1:
   \[
   2a + 4d = 40 \quad \Rightarrow \quad a + 2d = 20 \quad \text{(Equation 3)}
   \]

   From Equation 2:
   \[
   2a + 6d = 52 \quad \Rightarrow \quad a + 3d = 26 \quad \text{(Equation 4)}
   \]

   Subtract Equation 3 from Equation 4:
   \[
   (a + 3d) - (a + 2d) = 26 - 20
   \]
   \[
   d = 6
   \]

   Substitute $ d = 6 $ back into Equation 3:
   \[
   a + 2(6) = 20
   \]
   \[
   a + 12 = 20
   \]
   \[
   a = 8
   \]

   Therefore, the first term $ a $ is 8 and the common difference $ d $ is 6.

4. **Find the sum of the first ten terms:**

   The sum of the first $ n $ terms of an arithmetic sequence is given by:
   \[
   S_n = \frac{n}{2} (2a + (n-1)d)
   \]

   For $ n = 10 $:
   \[
   S_{10} = \frac{10}{2} (2 \cdot 8 + (10-1) \cdot 6)
   \]
   Simplify inside the parentheses:
   \[
   S_{10} = 5 (16 + 54)
   \]
   \[
   S_{10} = 5 \cdot 70
   \]
   \[
   S_{10} = 350
   \]

Thus, the sum of the first ten terms of the arithmetic sequence is $ \boxed{350} $.

Would you like more details on any specific part of the solution? Here are some additional questions that could expand on this topic:

1. How do you derive the formula for the sum of an arithmetic series?
2. What are some applications of arithmetic sequences in real life?
3. Can you explain how to solve a system of linear equations?
4. What is the general form of an arithmetic sequence?
5. How would the problem change if the sequence were geometric?
6. What is the significance of the common difference in an arithmetic sequence?
7. How can arithmetic sequences be used in financial calculations?
8. What are the limitations of using arithmetic sequences in modeling?

**Tip:** Always check your final answers by substituting back into the original equations to ensure consistency.

Learn how to find the sum of the first ten terms in an arithmetic sequence using given conditions. Detailed step-by-step solution involving arithmetic sequences and system of equations. Suitable for Grades 9-12.

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