How do you solve for the first term in an arithmetic sequence? Given a_3= -12, a_7= 8 

To solve for the first term (\(a_1\)) in an arithmetic sequence, we need to use the general formula for the \(n\)-th term of an arithmetic sequence:

\[
a_n = a_1 + (n - 1) \cdot d
\]

where:
- \(a_n\) is the \(n\)-th term.
- \(a_1\) is the first term.
- \(d\) is the common difference.
- \(n\) is the term number.

### Given Information
- \(a_3 = -12\)
- \(a_7 = 8\)

### Step-by-Step Solution

1. **Use the general formula for both terms:**

   For \(a_3\):
   \[
   a_3 = a_1 + (3 - 1) \cdot d = a_1 + 2d = -12
   \]

   For \(a_7\):
   \[
   a_7 = a_1 + (7 - 1) \cdot d = a_1 + 6d = 8
   \]

2. **Set up a system of equations:**

   From \(a_3 = -12\):
   \[
   a_1 + 2d = -12
   \]

   From \(a_7 = 8\):
   \[
   a_1 + 6d = 8
   \]

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

   Subtract the first equation from the second:

   \[
   (a_1 + 6d) - (a_1 + 2d) = 8 - (-12)
   \]

   This simplifies to:

   \[
   4d = 20 \implies d = 5
   \]

4. **Substitute the value of \(d\) back into one of the equations:**

   Using \(a_1 + 2d = -12\):

   \[
   a_1 + 2(5) = -12
   \]

   \[
   a_1 + 10 = -12 \implies a_1 = -22
   \]

### Conclusion

The first term of the arithmetic sequence is:

\[
\boxed{a_1 = -22}
\]

Would you like more details on how to solve similar problems? Here are a few related questions to consider:

1. How do you find the common difference in an arithmetic sequence?
2. What is the formula for the sum of the first \(n\) terms of an arithmetic sequence?
3. How do you determine if a given sequence is arithmetic?
4. How do you find the \(n\)-th term of an arithmetic sequence if you know the first term and common difference?
5. Can you solve problems involving arithmetic sequences with only partial information?

**Tip:** When solving problems involving arithmetic sequences, always write down the general formula and identify what information is given before solving.

Learn how to solve for the first term (a_1) in an arithmetic sequence using the given terms a_3 = -12 and a_7 = 8. Step-by-step solution provided, including the use of the general formula for the nth term of an arithmetic sequence. Suitable for high school students.

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