3.2 A linear number pattern with a constant difference can be represented by the terms \( x+3 ; 3 x+2 ; 6 x-1 \). Determine the numerical value \( x \) AND the numerical value the \( 3^{\text {rd }} \) term.

To determine the value of \( x \) in the linear number pattern represented by the terms \( x + 3 \), \( 3x + 2 \), and \( 6x - 1 \), we need to ensure that the difference between consecutive terms is constant. Let's denote the terms as follows: - First term: \( a_1 = x + 3 \) - Second term: \( a_2 = 3x + 2 \) - Third term: \( a_3 = 6x - 1 \) The difference between the second and first terms should equal the difference between the third and second terms: \[ a_2 - a_1 = a_3 - a_2 \] Calculating \( a_2 - a_1 \): \[ a_2 - a_1 = (3x + 2) - (x + 3) = 3x + 2 - x - 3 = 2x - 1 \] Calculating \( a_3 - a_2 \): \[ a_3 - a_2 = (6x - 1) - (3x + 2) = 6x - 1 - 3x - 2 = 3x - 3 \] Setting the two differences equal to each other: \[ 2x - 1 = 3x - 3 \] Now, we can solve for \( x \): \[ 2x - 1 = 3x - 3 \] \[ -1 + 3 = 3x - 2x \] \[ 2 = x \] Now that we have found \( x = 2 \), we can substitute this value back into the expression for the third term \( a_3 \): \[ a_3 = 6x - 1 = 6(2) - 1 = 12 - 1 = 11 \] Thus, the numerical value of \( x \) is \( 2 \) and the numerical value of the third term is \( 11 \). Final answers: - \( x = 2 \) - Third term \( a_3 = 11 \)