This work is concerned with the difference equation
\[ x_{n+k+1}=f_k(x_{n+k})\dots f_2(x_{n+2})f_1(x_{n+1}),\quad n\in \mathbb{N}_0:=\{0,1,2,\dots\},\quad k\in \mathbb{N},\quad k\geq 2,\tag{E} \]
where \(f_i:(0,\infty)\to (0,\infty)\), \(i=1,\dots, k\) are continuous functions. The authors prove that (E) is a \((k+1)\)-cycle if and only if it has the form \[ x_{n+k+1}=\displaystyle\frac{C}{x_{n+1}x_{n+2}\ldots x_{n+k}}\,, \]
for some positive constant \(C\), if \(k\) is an arbitrary natural number, or \[ x_{n+k+1}=\displaystyle\frac{\prod_{j=1}^{(k+1)/2}x_{n+2j-1}} {\prod_{j=1}^{(k-1)/2}x_{n+2j}}\,, \]
if \(k\) is an odd natural number.