The spin-1/2, quantum antiferromagnetic chains with linearly decreasing exchange coupling are constructed. The exchange coupling beyond a separation (R 1) is zero, where R is a positive integer. For odd values of R, the ground state of all the Hamiltonians consists of two degenerate singlet dimer configurations. These models have low-lying kink excitations with an energy gap. The Majumdar-Ghosh model is the first member of the odd family. The models with even values of R are qualitatively different from the ones with odd values, and admit an exact dimer ground state only in the asymptotic limit when R is infinitely large. The scheme of construction is applied in two and three dimensions, to construct quantum spin models with exact dimer ground states. The Shastry-Sutherland model in two dimensions is a well known example which fits in our general scheme. Many of these models have finite entropy density in their ground states. An interesting property of super stability is pointed out for the dimer ground states of these models.