This paper studies the behavior of positive solutions of the recursive equation
y
n
=
A
+
(
y
n
−
k
y
n
−
m
)
p
,
n
=
0
,
1
,
2
,
…
,
\begin{eqnarray} y_n=A+\left (\frac {y_{n-k}}{y_{n-m}}\right )^p,\quad n=0,1,2,\ldots , \nonumber \end{eqnarray}
with
y
−
s
,
y
−
s
+
1
,
…
,
y
−
1
∈
(
0
,
∞
)
y_{-s},y_{-s+1}, \ldots , y_{-1} \in (0, \infty )
and
k
,
m
∈
{
1
,
2
,
3
,
4
,
…
}
k,m \in \{1,2,3,4,\ldots \}
, where
s
=
max
{
k
,
m
}
s=\max \{k,m\}
. We prove that if
g
c
d
(
k
,
m
)
=
1
\mathrm {gcd}(k,m) = 1
, and
p
≤
min
{
1
,
(
A
+
1
)
/
2
}
p\leq \min \{1,(A+1)/2\}
, then
y
n
y_n
tends to
A
+
1
A+1
. This complements several results in the recent literature, including the main result in K. S. Berenhaut, J. D. Foley and S. Stević, The global attractivity of the rational difference equation
y
n
=
1
+
y
n
−
k
y
n
−
m
y_{n}=1+\frac {y_{n-k}}{y_{n-m}}
, Proc. Amer. Math. Soc., 135 (2007) 1133–1140.