The calculated formula for partition function of one-Dimensional Fibonacci quasiperiodic frustrated Ising system with free and periodic boundary conditions are given in this paper. Its thermodynamical properties at low temperatures are studied. It is found that when the temperature tends to zero , the thermodynamical quantities as the functions of B/J show discontinuities at B/J=2/(2m+l) (m interger). The ground state configuration of system in various parameter regions are described and the results of calculation are discussed.
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