2.1 Determine which term of the sequence \( 23 ; 21 ; 19 ; \ldots \)

To determine which term of the sequence \( 23, 21, 19, \ldots \) corresponds to a specific value, we first need to identify the pattern of the sequence. 1. **Identify the pattern**: The sequence starts at 23 and decreases by 2 each time: - \( 23 \) (1st term) - \( 21 \) (2nd term) - \( 19 \) (3rd term) - and so on. 2. **General formula for the \( n \)-th term**: The \( n \)-th term of an arithmetic sequence can be expressed as: \[ a_n = a_1 + (n - 1) \cdot d \] where: - \( a_1 \) is the first term, - \( d \) is the common difference, - \( n \) is the term number. In this case: - \( a_1 = 23 \) - \( d = -2 \) Therefore, the formula for the \( n \)-th term becomes: \[ a_n = 23 + (n - 1)(-2) \] Simplifying this gives: \[ a_n = 23 - 2(n - 1) = 23 - 2n + 2 = 25 - 2n \] 3. **Finding a specific term**: If we want to find which term corresponds to a specific value, we can set \( a_n \) equal to that value and solve for \( n \). For example, if we want to find out which term is \( 19 \): \[ 19 = 25 - 2n \] Rearranging gives: \[ 2n = 25 - 19 \] \[ 2n = 6 \] \[ n = 3 \] Thus, the term \( 19 \) is the 3rd term of the sequence. If you have a specific value in mind that you want to check, please provide it, and I can determine which term corresponds to that value.