Exercise 3, § 1

Let $K$ be a field, $B$ the polynomial ring $K[X, Y]$ with the graduation defined by the total degree, $C$ the graded subring of $B$ generated by $X$ and $Y^2$ and $a$ the graded ideal of $C$ generated by $Y^4$. Show that, in the graded ring $A = C/a$, $A_{n+1} = A_1 A_n$ for $n \geq 2$ but $A_n \neq (A_1)^n$ for $n \geq 6$.