In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. The $n$-th term of an arithmetic sequence can be expressed as:

$a_n = a + (n-1)d$

where:

• $a_n$ is the $n$-th term,
• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the term number.

Given:

• The 5th term ($a_5$) is 22,
• The 12th term ($a_{12}$) is 51.

We can write the equations for these terms as:

$a_5 = a + 4d = 22$ $a_{12} = a + 11d = 51$

Now we have a system of two linear equations:

1. $a + 4d = 22$
2. $a + 11d = 51$

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

$(a + 11d) - (a + 4d) = 51 - 22$ $a + 11d - a - 4d = 29$ $7d = 29$ $d = \frac{29}{7} = 4.142857142857143$

Now that we have $d$, we can substitute it back into the first equation to find $a$:

$a + 4 \left(\frac{29}{7}\right) = 22$ $a + \frac{116}{7} = 22$ $a = 22 - \frac{116}{7}$ $a = \frac{154}{7} - \frac{116}{7}$ $a = \frac{38}{7} = 5.428571428571429$

So, the first term $a$ is approximately 5.43, and the common difference $d$ is approximately 4.14.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. What is the general form of the $n$-th term in an arithmetic sequence?
2. How do you find the sum of the first $n$ terms in an arithmetic sequence?
3. What are the conditions for a sequence to be classified as arithmetic?
4. How would you determine the number of terms in an arithmetic sequence given the first term, common difference, and a specific term value?
5. Can an arithmetic sequence have a common difference of zero? What would that imply?
6. How would you solve for the common difference if given the first and last terms of a sequence, along with the total number of terms?
7. How can you determine if a given set of numbers forms an arithmetic sequence?
8. What are some real-world examples where arithmetic sequences are applicable?

Tip: When solving systems of linear equations, it's often helpful to use substitution or elimination methods to simplify the problem.