Let K be an infinite field and let m 1 ⋯ m n be a generalized arithmetic sequence of positive integers, i.e., there exist h , d , m 1 ∈ Z + such that m i = h m 1 + ( i − 1 ) d for all i ∈ { 2 , … , n } . We consider the projective monomial curve C ⊂ P K n parametrically defined by x 1 = s m 1 t m n − m 1 , … , x n − 1 = s m n − 1 t m n − m n − 1 , x n = s m n , x n + 1 = t m n . In this work, we characterize the Cohen–Macaulay and Koszul properties of the homogeneous coordinate ring K [ C ] of C . Whenever K [ C ] is Cohen–Macaulay we also obtain a formula for its Cohen–Macaulay type. Moreover, when h divides d , we obtain a minimal Grobner basis G of the vanishing ideal of C with respect to the degree reverse lexicographic order. From G we derive formulas for the Castelnuovo–Mumford regularity, the Hilbert series and the Hilbert function of K [ C ] in terms of the sequence.