Given the linear pattern: \( x ; y ; 83 ; 80 ; \)...... 3.2 .1 Calculate the values of \( x \) and \( y \). 3.2 .2 If \( x=89 \) and \( y=86 \), determine the general term of this pattern. 3.2 .3 Calculate the value of \( T_{1000} \).

Let's solve each part of the problem step by step. --- ### **3.2.1 Calculate the values of \( x \) and \( y \).** **Given Sequence:** \( x ; y ; 83 ; 80 ; \ldots \) Since the pattern is linear, we can assume it's an **arithmetic sequence**, where the difference between consecutive terms is constant. **Let the common difference be \( d \).** 1. **Set up the equations based on the common difference:** - \( y - x = d \)   (1) - \( 83 - y = d \)   (2) - \( 80 - 83 = d \)   (3) 2. **Solve equation (3) to find \( d \):** \[ 80 - 83 = d \\ d = -3 \] 3. **Substitute \( d = -3 \) into equation (2) to find \( y \):** \[ 83 - y = -3 \\ y = 83 + 3 \\ y = 86 \] 4. **Substitute \( d = -3 \) and \( y = 86 \) into equation (1) to find \( x \):** \[ 86 - x = -3 \\ x = 86 + 3 \\ x = 89 \] **Answer:** \( x = 89 \) and \( y = 86 \). --- ### **3.2.2 Determine the general term of this pattern if \( x = 89 \) and \( y = 86 \).** **Given:** - First term, \( a_1 = 89 \) - Common difference, \( d = -3 \) **General Formula for an Arithmetic Sequence:** \[ a_n = a_1 + (n - 1) \times d \] **Substitute the known values:** \[ a_n = 89 + (n - 1)(-3) \\ a_n = 89 - 3(n - 1) \\ a_n = 89 - 3n + 3 \\ a_n = 92 - 3n \] **Answer:** The general term is \( a_n = 92 - 3n \). --- ### **3.2.3 Calculate the value of \( T_{1000} \).** **Using the general term derived:** \[ a_n = 92 - 3n \] **Find \( a_{1000} \):** \[ a_{1000} = 92 - 3 \times 1000 \\ a_{1000} = 92 - 3000 \\ a_{1000} = -2908 \] **Answer:** \( T_{1000} = -2908 \). ---