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.