QUESTION 3 3.1 Given the linear pattern: \( 7 ; 2 ;-3 ; \ldots \) 3.1.1 \( \quad \) Determine the general term, \( T_{n} \), of the linear pattern. \( \begin{array}{l}\text { 3.1.2 } \\ \begin{array}{l}\text { 3.1.3 } \\ \\ 3.2\end{array} \quad \text { Which term in the pattern has a value of }-138 \text { ? } \\ \text { Calculate the value of } x\end{array} \)

To solve the problem, we will follow these steps: ### Step 1: Determine the general term \( T_n \) of the linear pattern The given linear pattern is: \( 7, 2, -3, \ldots \) 1. **Identify the first term**: - \( T_1 = 7 \) 2. **Identify the common difference**: - The difference between the first term and the second term is \( 2 - 7 = -5 \). - The difference between the second term and the third term is \( -3 - 2 = -5 \). - Thus, the common difference \( d = -5 \). 3. **Use the formula for the \( n \)-th term of an arithmetic sequence**: \[ T_n = T_1 + (n - 1) \cdot d \] Substituting the known values: \[ T_n = 7 + (n - 1)(-5) \] Simplifying this: \[ T_n = 7 - 5(n - 1) \] \[ T_n = 7 - 5n + 5 \] \[ T_n = 12 - 5n \] ### Step 2: Find which term in the pattern has a value of -138 We need to solve for \( n \) when \( T_n = -138 \): \[ 12 - 5n = -138 \] Now, we will solve this equation step by step: 1. **Rearranging the equation**: \[ -5n = -138 - 12 \] \[ -5n = -150 \] 2. **Dividing both sides by -5**: \[ n = \frac{-150}{-5} = 30 \] ### Final Answers 1. The general term \( T_n \) of the linear pattern is: \[ T_n = 12 - 5n \] 2. The term in the pattern that has a value of -138 is the 30th term: \[ n = 30 \]