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We consider the following nonlinear difference equation: xn+1=f(pn,xn−m,xn−t(k+1)+1), n=0,1,2,…, where m∈{0,1,2,…} and k,t∈{1,2,…} with 0≤m<t(k+1)−1, the initial values x−t(k+1)+1,x−t(k+1)+2,…,x0∈(0,+∞), and {pn}n=0∞ is a positive sequence of the period k+1. We give sufficient conditions under which every positive solution of this equation tends to the period k+1 solution.