The Grothendieck groups of n-cluster categories of type A_{\infty}

In this article, we investigate the Grothendieck groups $K_0({\text{C}}_{A_{\mathrm{\infty }}}^n)$ of $n$-cluster categories ${\text{C}}_{A_{\mathrm{\infty }}}^n$ of type $A_{\mathrm{\infty }}$ introduced by T.~Holm and P.~Jørgensen. We prove that $K_0({\text{C}}_{A_{\mathrm{\infty }}}^n)\cong \mathbb{Z}$ for an arbitrary $n\ge 1$. As an application, this generalizes a result of Murphy for $n=1$.