Write an arithmetic formula, if possible, to describe the sequence of numbers and determine \[ 7,12,17,22,27 \ldots \] \( a_{8}=5+7(n-1) ; a_{8}=54 \) \( a_{n}=7-5(n-1) ; a_{8}=-28 \) \( a_{n}=7+5(n-1) ; a_{8}=42 \) \( a_{n}=5-7(n-1) ; a_{8}=-44 \)

To describe the sequence \( 7, 12, 17, 22, 27, \ldots \), we first observe the pattern in the numbers. 1. **Identify the pattern:** - The first term \( a_1 = 7 \) - The second term \( a_2 = 12 \) - The third term \( a_3 = 17 \) - The fourth term \( a_4 = 22 \) - The fifth term \( a_5 = 27 \) We can see that each term increases by 5. Therefore, this is an arithmetic sequence with: - First term \( a_1 = 7 \) - Common difference \( d = 5 \) 2. **General formula for the \( n \)-th term:** The formula for the \( n \)-th term of an arithmetic sequence can be expressed as: \[ a_n = a_1 + (n-1) \cdot d \] Substituting the known values: \[ a_n = 7 + (n-1) \cdot 5 \] 3. **Simplifying the formula:** \[ a_n = 7 + 5n - 5 = 5n + 2 \] Now, we can check the provided options to see which one matches our derived formula. - **Option 1:** \( a_{8}=5+7(n-1) \) - **Option 2:** \( a_{8}=7-5(n-1) \) - **Option 3:** \( a_{8}=7+5(n-1) \) - **Option 4:** \( a_{8}=5-7(n-1) \) 4. **Evaluate the options:** - For **Option 1**: \( a_n = 5 + 7(n-1) \) does not match. - For **Option 2**: \( a_n = 7 - 5(n-1) \) does not match. - For **Option 3**: \( a_n = 7 + 5(n-1) \) simplifies to \( a_n = 5n + 2 \), which matches our derived formula. - For **Option 4**: \( a_n = 5 - 7(n-1) \) does not match. 5. **Conclusion:** The correct formula for the sequence is: \[ a_n = 7 + 5(n-1) \] Thus, the correct option is **Option 3**. Now, let's calculate \( a_8 \) using our derived formula: \[ a_8 = 7 + 5(8-1) = 7 + 5 \cdot 7 = 7 + 35 = 42 \] This confirms that \( a_8 = 42 \).