Consider the tinite linear pattern: \( 20 ; 17 ; 14 ; \ldots ;-103 \) 3.1 Wetermine the expression for the \( n^{\text {in }} \) term. 3.2 Calculate the number of terms in the sequence. 3.4 Which term is the first to have a negative value? 3.5 What is the value of the \( 19^{\text {th }} \) even-valued term in the sequence?

To analyze the finite linear pattern \( 20, 17, 14, \ldots, -103 \), we can identify the first term and the common difference. ### 3.1 Determine the expression for the \( n^{\text{th}} \) term. The first term \( a_1 \) is \( 20 \) and the common difference \( d \) can be calculated as follows: \[ d = 17 - 20 = -3 \] The general formula for the \( n^{\text{th}} \) term of an arithmetic sequence is given by: \[ a_n = a_1 + (n-1) \cdot d \] Substituting the values we have: \[ a_n = 20 + (n-1)(-3) \] \[ a_n = 20 - 3(n-1) \] \[ a_n = 20 - 3n + 3 \] \[ a_n = 23 - 3n \] ### 3.2 Calculate the number of terms in the sequence. To find the number of terms, we need to determine \( n \) when \( a_n = -103 \): \[ 23 - 3n = -103 \] Solving for \( n \): \[ 23 + 103 = 3n \] \[ 126 = 3n \] \[ n = \frac{126}{3} = 42 \] Thus, there are **42 terms** in the sequence. ### 3.4 Which term is the first to have a negative value? To find the first negative term, we need to find the smallest \( n \) such that \( a_n < 0 \): \[ 23 - 3n < 0 \] \[ 23 < 3n \] \[ \frac{23}{3} < n \] \[ n > 7.67 \] Since \( n \) must be a whole number, the smallest integer \( n \) that satisfies this inequality is \( n = 8 \). Thus, the **first term to have a negative value is the 8th term**. ### 3.5 What is the value of the \( 19^{\text{th}} \) even-valued term in the sequence? To find the even-valued terms, we need to identify the terms in the sequence that are even. The general term is: \[ a_n = 23 - 3n \] We want \( a_n \) to be even: \[ 23 - 3n \text{ is even} \] Since \( 23 \) is odd, \( -3n \) must also be odd, which means \( n \) must be odd (because the product of an odd number and an odd number is odd). The odd integers are \( 1, 3, 5, 7, \ldots \). The \( k^{\text{th}} \) odd integer can be expressed as \( n = 2k - 1 \). To find the \( 19^{\text{th}} \) even-valued term, we set \( k = 19 \): \[ n = 2(19) - 1 = 37 \] Now we calculate \( a_{37} \): \[ a_{37} = 23 - 3(37) \] \[ a_{37} = 23 - 111 \] \[ a_{37} = -88 \] Thus, the value of the \( 19^{\text{th}} \) even-valued term in the sequence is **-88**.