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 (\(a\)) and the common difference (\(d\)) of an arithmetic sequence, we can use the information given about the 5th and 12th terms. 

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

We know:
- The 5th term (\(a_5\)) is 22.
- The 12th term (\(a_{12}\)) is 51.

Using the formula for the 5th term:
\[ a_5 = a + 4d = 22 \]

Using the formula for the 12th term:
\[ a_{12} = a + 11d = 51 \]

We now have the following system of linear equations:
1. \( a + 4d = 22 \)
2. \( a + 11d = 51 \)

To solve this system, we can subtract the first equation from the second equation:
\[ (a + 11d) - (a + 4d) = 51 - 22 \]
\[ a + 11d - a - 4d = 29 \]
\[ 7d = 29 \]
\[ d = \frac{29}{7} \]
\[ d = 4.14 \]

Next, we substitute \(d\) back into the first equation to find \(a\):
\[ a + 4(4.14) = 22 \]
\[ a + 16.56 = 22 \]
\[ a = 22 - 16.56 \]
\[ a = 5.44 \]

Therefore, the first term and the common difference of the arithmetic sequence are:
\[ a = 5.44 \]
\[ d = 4.14 \]

Would you like more details or have any questions?

Here are 8 related questions to expand on this topic:
1. How do you derive the formula for the \(n\)-th term of an arithmetic sequence?
2. How can you find the sum of the first \(n\) terms of an arithmetic sequence?
3. What happens to an arithmetic sequence if the common difference is negative?
4. How would you find the 20th term of this arithmetic sequence?
5. How can you determine if a given sequence is arithmetic?
6. What are the properties of arithmetic sequences?
7. Can an arithmetic sequence also be a geometric sequence? Why or why not?
8. How would you find the common difference if only the first and last terms are given along with the number of terms?

**Tip:** In solving systems of linear equations, checking your work by substituting the values back into the original equations can help ensure accuracy.

Learn how to find the first term and common difference of an arithmetic sequence using the 5th and 12th terms provided. This problem involves solving a system of linear equations to determine the parameters of the sequence. Suitable for students in Grades 9-12.

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