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:

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.