Determine the Given system is linear or nonlinear
Using the definition of linear system as one which can be put in the form:

$y_0 + a_1 y_1 + \dots + a_n y_n = b_0 x_0 + \dots + b_m x_m$

because of the multiplicative term $y(n)y(2n)$ we can conclude that the system is nonlinear.