The 5th term of an arithmetic sequence is 22 and the 12th term is 51. Find the first term and the common difference.

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.

Learn how to find the first term and common difference in an arithmetic sequence with detailed step-by-step solution. This problem involves solving a system of linear equations derived from specific terms in the sequence.

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