3.1 Given the linear pattern: \( 7 ; 2 ;-3 ; \ldots \) 3.1.1 Determine the general term, \( T_{n} \), of the linear pattern. 3.1.2 Calculate the value of \( T_{20} \). 3.1.3 Which term in the pattern has a value of -138 ?

**Step 1. Find the general term \( T_n \) for the sequence** The pattern is given by the terms: \[ 7,\quad 2,\quad -3,\quad \ldots \] The common difference \( d \) is calculated as: \[ d = 2 - 7 = -5 \] A linear (arithmetic) sequence has the general term: \[ T_n = a + (n-1)d \] where \( a \) is the first term. Substituting \( a = 7 \) and \( d = -5 \): \[ T_n = 7 + (n-1)(-5) \] Expanding the expression: \[ T_n = 7 - 5(n-1) \] \[ T_n = 7 - 5n + 5 \] \[ T_n = 12 - 5n \] --- **Step 2. Calculate \( T_{20} \)** Using the general formula \( T_n = 12 - 5n \), substitute \( n = 20 \): \[ T_{20} = 12 - 5(20) \] \[ T_{20} = 12 - 100 \] \[ T_{20} = -88 \] --- **Step 3. Determine the term number for which the value is \(-138\)** We need to find \( n \) such that: \[ T_n = -138 \] Substitute the formula \( T_n = 12 - 5n \): \[ 12 - 5n = -138 \] Subtract 12 from both sides: \[ -5n = -138 - 12 \] \[ -5n = -150 \] Divide by \(-5\): \[ n = \frac{-150}{-5} \] \[ n = 30 \] --- **Final Answers:** 1. The general term is: \[ T_n = 12 - 5n \] 2. The 20th term is: \[ T_{20} = -88 \] 3. The term with the value \(-138\) is the 30th term.