\( x+3 ; 2 x+6 ; 3 x+9 ; \ldots \) are the first three terms of an arithmetic sequence. (a) Determine the 10 th term in terms of \( x \). (b) Determine the \( n \)th term in terms of \( x \).

The given terms of the sequence are \( x+3 \), \( 2x+6 \), and \( 3x+9 \). To show that these terms form an arithmetic sequence, we need to confirm that the difference between consecutive terms is constant. The common difference is: \[ (2x+6) - (x+3) = x + 3 \] \[ (3x+9) - (2x+6) = x + 3 \] Since both differences are equal, the sequence is indeed arithmetic. (a) To find the 10th term, we can use the formula for the nth term of an arithmetic sequence: \[ T_n = T_1 + (n-1)d, \] where \( T_1 = x + 3 \) and \( d = x + 3 \). Plugging \( n=10 \) into the formula gives us: \[ T_{10} = (x + 3) + (10 - 1)(x + 3) = (x + 3) + 9(x + 3) = 10(x + 3) = 10x + 30. \] (b) The nth term can be expressed as: \[ T_n = T_1 + (n-1)d = (x + 3) + (n - 1)(x + 3) = (x + 3) + (n-1)(x + 3) = nx + 3n. \] Thus, the nth term of the sequence is: \[ T_n = nx + 3n. \]