This study proves a general result on convergence of α ˜ 2 x ∈ Ext s+2, tq+2q+1 A( Z p , Z p ) in the Adams spectral sequence and as a consequence, the study detects some new families in the stable homotopy groups of spheres π tq+2q−4S which is represented in the Adams spectral sequence by α ˜ 2f n, α ˜ 2f′ n, α ˜ 2h uh mh n ∈ Ext 5, tq+2q+1 A( Z p , Z p ) with tq = p n+1 q+2 p n q, 2 p n+1 q+p n q, p u q+p m q+p n q, respectively, where α ˜ 2 ∈ Ext 2,2 q+1 a ( Z p , Z p ), f n ∈ Ext 3, p n+1 q+2p nq A ( Z p , Z p ), f′ n ∈ Ext 3,2 p n+1 q+p nq A ( Z p , Z p ), h n ∈ Ext 1, p nq A ( Z p , Z p ) and p ≥ 5 is a prime, q = 2( p − 1), n ≥ 2.