Summary: This paper computes the spectrum of the forward difference operator \(\Delta^{r+}_\nu\) of order \(r\in \mathbb{N}\) over the Banach space \(\ell_1\). Indeed, the difference operator \(\Delta_\nu^{r+}\) via the sequence \(x=(x_k)\) is defined by \[ (\Delta_\nu^{r+}x)_k = (\Delta_\nu^{(r-1)+} x)_k- (\Delta_\nu^{(r-1)+} x)_{k+1}=\sum_{i=0}^r(-1)^i\binom{r}{i} \nu_{k+1}x_{k+i}, \] where \(x = (x_k)\in\ell_1\), \(r\in \mathbb{N}\) and \(\nu = (\nu_k)\) is either a constant or strictly decreasing sequence of positive real numbers with some conditions. Subdivision sets of the spectrum such as the point, the residual and the continuous spectrum of the operator \(\Delta_\nu^{r+}\) on the Banach space \(\ell_1\) are computed. In this context, some illustrative examples are provided.