A linear composition of a positive integer n is a finite sequence of positive integers (called parts) whose sum equals n. A cyclic composition of n is an equivalent class of all linear compositions of n that can be obtained from each other by a cyclic shift. In this paper, we enumerate the cyclic compositions of n that avoid an increasing arithmetic sequence of positive integers. In the case where all multiples of a positive integer r are avoided, we show that the number of cyclic compositions of n with this property equals to or is one less than the number of cyclic zero-one sequences of length n that do not contain r consecutive ones. In addition, we show that this number is related to the r-step Lucas numbers.