In this paper, we investigate the asymptotic stability of the recursive sequence x n + 1 = α + β x n 2 1 + γ x n − 1 ,         n = 0 , 1 , … and the existence of certain monotonic solutions of the equation x n + 1 = x n p f ( x n , x n − 1 , … , x n − k ) ,           n = 0 , 1 , … which includes as a special case the rational recursive sequence x n + 1 = β x n p 1 + ∑ i = 1 k γ i x n − 1 p − r , where 0,\gamma > 0,\gamma _i \geqslant 0 \]"> α ≥ 0 , β > 0 , γ > 0 , γ i ≥ 0 , 0} \]"> i = 1 , 2 , … , k , ∑ i = 1 k γ i > 0 , p ∈ { 2 , 3 , … } and r ∈ { 1 , 2 , … , p − 1 } . The case when r = 0 has been investigated by Camouzis et. al. [1], and for r = 0 and p = 2 by Camouzis et. al. [2].