Find the number of terms in the arithmetic sequence \( -2 ;-6 ;-10 ; \ldots ;-150 \).

To find the number of terms in the arithmetic sequence given, we first identify the first term, common difference, and the last term. The first term \( a \) is given as: \[ a = -2 \] The common difference \( d \) can be calculated by subtracting the first term from the second term: \[ d = -6 - (-2) = -6 + 2 = -4 \] The last term we need to reach in the sequence is: \[ l = -150 \] The general formula for the \( n \)-th term of an arithmetic sequence is: \[ a_n = a + (n - 1) d \] Setting \( a_n \) equal to the last term \( l \), we have: \[ -150 = -2 + (n - 1)(-4) \] We can simplify this equation step by step. First, add 2 to both sides: \[ -150 + 2 = (n - 1)(-4) \] \[ -148 = (n - 1)(-4) \] Next, divide both sides by \(-4\): \[ \frac{-148}{-4} = n - 1 \] \[ 37 = n - 1 \] Finally, add 1 to both sides to solve for \( n \): \[ n = 37 + 1 = 38 \] Thus, the total number of terms in the arithmetic sequence is: \[ \boxed{38} \]