The sum of the interior angles of a hexagon is 180(6−2)=720 degrees. Let the smallest angle in the arithmetic sequence be x degrees. Then the other angles in the sequence are x+d,x+2d,…,x+5d degrees, where d is the common difference. Since the sum of the angles is 720 degrees, we have the equation [x + (x + d) + (x + 2d) + \dots + (x + 5d) = 720.]This is an arithmetic series with first term x and common difference d, so its sum is [\frac{x + (x + 5d)}{2} \cdot 6 = 3x + 15d = 720.]Then x+5d=240. Since x and d are positive integers less than 150, the only possible values of x are 120, 121, 122, …, 148. For each of these values of x, there is exactly one corresponding value of d, namely 240−x. Therefore, there are 29 possible sequences.
