Consider the arithmetic sequence \( 12: 8 ; 4 ; \ldots \) (1) Calculate \( T_{23} \). (2) Determine the value of \( k \) if \( T_{2}=-168 \)

Let's analyze the given arithmetic sequence and solve both parts step by step. ### Understanding the Sequence An **arithmetic sequence** is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the **common difference** (\( d \)). The given sequence is: \[ 12, \, 8, \, 4, \, \ldots \] From this, we can identify: - **First term (\( a \))**: \( 12 \) - **Second term**: \( 8 \) - **Third term**: \( 4 \) ### Finding the Common Difference (\( d \)) \[ d = \text{Second term} - \text{First term} = 8 - 12 = -4 \] Thus, the common difference \( d = -4 \). ### (1) Calculating \( T_{23} \) The \( n \)-th term of an arithmetic sequence is given by: \[ T_n = a + (n - 1) \times d \] Substituting the known values: \[ T_{23} = 12 + (23 - 1) \times (-4) \] \[ T_{23} = 12 + 22 \times (-4) \] \[ T_{23} = 12 - 88 \] \[ T_{23} = -76 \] ### (2) Determining the Value of \( k \) if \( T_2 = -168 \) Assuming \( k \) represents the **common difference** in this part, we can set up the equation using the formula for the \( n \)-th term: \[ T_2 = a + (2 - 1) \times d = a + d \] Given: \[ T_2 = -168 \] \[ a = 12 \] Substitute the known values: \[ -168 = 12 + d \] \[ d = -168 - 12 \] \[ d = -180 \] Thus, the value of \( k \) (the common difference) is: \[ k = -180 \] ### Summary of Answers 1. **\( T_{23} = -76 \)** 2. **\( k = -180 \)**