Given the information about the terms of an arithmetic sequence, we can find the first term and the common difference of the sequence.

Let's denote:

• $a$ as the first term of the arithmetic sequence.
• $d$ as the common difference.
• $a_n$ as the nth term of the arithmetic sequence.

The general formula for the nth term of an arithmetic sequence is: $a_n = a + (n - 1)d$

Given:

1. The 8th term ($a_8$) is 12: $a + 7d = 12$
2. The 12th term ($a_{12}$) is 8: $a + 11d = 8$

We now have the following system of linear equations: $a + 7d = 12$ $a + 11d = 8$

To solve for $a$ and $d$, we can subtract the first equation from the second equation:

$(a + 11d) - (a + 7d) = 8 - 12$ $4d = -4$ $d = -1$

Now that we have $d$, we can substitute it back into one of the original equations to find $a$:

$a + 7(-1) = 12$ $a - 7 = 12$ $a = 19$

Therefore, the first term $a$ is 19, and the common difference $d$ is -1.

Summary:

• First term ($a$) = 19
• Common difference ($d$) = -1

Would you like more details or have any questions about this solution?

Here are some related questions you might want to ask:

1. How do you find the nth term of an arithmetic sequence?
2. How do you determine the common difference in an arithmetic sequence?
3. What is the sum of the first n terms in an arithmetic sequence?
4. How do you solve systems of linear equations?
5. What are the characteristics of arithmetic sequences compared to geometric sequences?

Tip: When dealing with arithmetic sequences, always start by identifying the first term and the common difference, as these are the building blocks for all other calculations.