If $I$ is generated by a regular sequence, $bigoplus_{n geq 0}I^{n}/I^{n+1}$ is isomorphic to a polynomial ring

Let $R$ be a Cohen-Macaulay ring and $I$ be an ideal generated by a regular sequence. I want to show that:

$bigoplus_{n geq 0}I^{n}/I^{n+1}$ is isomorphic to a polynomial ring
over $R/I$ in as many variables as generators of $I$.

My attempt: Let $S$ be the polynomial ring $(R/I)[x_{0},…,x_{n}]$. Then we have $S=bigoplus_{d geq 0}S_d$, which $S_d$ is the set of linear combinations of monomials with total weight $d$.
I tried to show that there is an isomorphism between $I^{n}/I^{n+1}$ and $S_{n}$ for $n geq 0$. Let $(a_{0},…,a_{m})$ be the regular sequence by which $I$ is generated.
Consider $beta : I^{n}/I^{n+1} rightarrow S_n$ with $beta(r.a_{i_1}…a_{i_n}+I^{n+1})= (r+I).x_{i_1}…x_{i_n}. $ I’m not sure if it works since I haven’t used the fact that $(a_{0},…,a_{m})$ is a regular sequence.