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

To find the first term and the common difference of an arithmetic sequence where the 5th term is 22 and the 12th term is 51, we follow these steps:

1. Use the general formula for the \(n\)-th term of an arithmetic sequence:
\[ a_n = a + (n-1)d \]

Given:
- The 5th term \(a_5 = 22\)
- The 12th term \(a_{12} = 51\)

2. Write the equations for these terms:

For the 5th term:
\[ a + 4d = 22 \]

For the 12th term:
\[ a + 11d = 51 \]

3. Set up a system of equations:

\[ \begin{cases}
a + 4d = 22 \\
a + 11d = 51
\end{cases} \]

4. Subtract the first equation from the second to eliminate \(a\):

\[ (a + 11d) - (a + 4d) = 51 - 22 \]
\[ a + 11d - a - 4d = 29 \]
\[ 7d = 29 \]
\[ d = \frac{29}{7} = 4.14 \]

5. Substitute the value of \(d\) back into the first equation to find \(a\):

\[ a + 4 \left(\frac{29}{7}\right) = 22 \]
\[ a + \frac{116}{7} = 22 \]
\[ a = 22 - \frac{116}{7} \]
\[ a = \frac{154}{7} - \frac{116}{7} \]
\[ a = \frac{38}{7} = 5.43 \]

Therefore, 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 of an arithmetic sequence with specific terms given. Step-by-step solution provided with detailed calculations and explanations. Suitable for high school students studying arithmetic sequences.

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