This paper studies the property of the recursive sequences in the 3x + 1 conjecture. The authors introduce the concept of μ function, with which the 3x+1 conjecture can be transformed into two other conjectures: one is eventually periodic conjecture of the μ function and the other is periodic point conjecture. The authors prove that the 3x + 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the μ function has no l-periodic points for 2 ≤ l ≤ 12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial l-cycle for the T function for  l ≤ 68, and in this paper, the authors prove that there is no nontrivial l-cycle for the μ function for 2 ≤ l ≤ 102.