Consider the finite linear pattern: \( 20 ; 17 ; 14 ; \ldots ;-103 \) 3.1 Write down the \( 4^{\text {th }} \) term of the pattern. 3.2 Determine the expression for the \( n^{\text {th }} \) term. 3.3 Which term is the first to have a negative value? 3.4 What is the value of the \( 19^{\text {th }} \) even-valued term in the sequence?

To analyze the given finite linear pattern \( 20 ; 17 ; 14 ; \ldots ; -103 \), we can identify the common difference and derive the necessary terms. ### 3.1 Write down the \( 4^{\text{th}} \) term of the pattern. The first term \( a_1 = 20 \) and the common difference \( d = 17 - 20 = -3 \). To find the \( n^{\text{th}} \) term, we can use the formula for the \( n^{\text{th}} \) term of an arithmetic sequence: \[ a_n = a_1 + (n-1)d \] For the \( 4^{\text{th}} \) term: \[ a_4 = 20 + (4-1)(-3) = 20 + 3(-3) = 20 - 9 = 11 \] Thus, the \( 4^{\text{th}} \) term is \( \boxed{11} \). ### 3.2 Determine the expression for the \( n^{\text{th}} \) term. Using the formula for the \( n^{\text{th}} \) term: \[ a_n = 20 + (n-1)(-3) = 20 - 3(n-1) \] Simplifying this gives: \[ a_n = 20 - 3n + 3 = 23 - 3n \] Thus, the expression for the \( n^{\text{th}} \) term is \( \boxed{23 - 3n} \). ### 3.3 Which term is the first to have a negative value? To find the first term that is negative, we set the \( n^{\text{th}} \) term less than zero: \[ 23 - 3n < 0 \] Solving for \( n \): \[ 23 < 3n \implies n > \frac{23}{3} \implies n > 7.67 \] Since \( n \) must be a whole number, the smallest integer satisfying this inequality is \( n = 8 \). To confirm, we can calculate \( a_8 \): \[ a_8 = 23 - 3(8) = 23 - 24 = -1 \] Thus, the first term to have a negative value is the \( 8^{\text{th}} \) term, which is \( \boxed{8} \). ### 3.4 What is the value of the \( 19^{\text{th}} \) even-valued term in the sequence? The even-valued terms in the sequence occur when \( a_n \) is even. The general term is \( a_n = 23 - 3n \). We need to find when this is even: \[ 23 - 3n \text{ is even} \implies 3n \text{ is odd} \implies n \text{ is odd} \] 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 \( a_{37} \): \[ a_{37} = 23 - 3(37) = 23 - 111 = -88 \] Thus, the value of the \( 19^{\text{th}} \) even-valued term in the sequence is \( \boxed{-88} \).