On the Recursive Sequencexn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj

We give a complete picture regarding the behavior of positive solutions of the following important difference equation:xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj,n∈ℕ0, whereαi, i∈{1,…,k}, andβj, j∈{1,…,m}, are positive numbers such that∑i=1kαi=∑j=1mβj=1,andpi, i∈{1,…,k}, andqj, j∈{1,…,m}, are natural numbers such thatp1<p2<⋯<pkandq1<q2<⋯<qm. The case whengcd(p1,…,pk,q1,…,qm)=1is the most important. For the case we prove that if allpi, i∈{1,…,k}, are even and allqj, j∈{1,…,m}, are odd, then every positive solution of this equation converges to a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.