If the system is isolated, then the center of mass of this system moves at a constant (usually zero) velocity $\mathbf{v}_c$:
$$
\sum_{i=1}^{N+1}m_i\mathbf{r}_i(t)=M \mathbf{r}_c(t)=M(\mathbf{r}_0+\mathbf{v}_c t)
$$
If $\mathbf{r}_i(t)$ are known for all $i=1,\ldots,N$, then $\mathbf{r}_{N+1}(t)$ can be obtained from this equation:
\begin{equation}\tag{1}
\mathbf{r}_{N+1}(t)=\frac{1}{m_{N+1}}\left(M(\mathbf{r}_0+\mathbf{v}_c t)-\sum_{i=1}^{N}m_i\mathbf{r}_i(t)\right)
\end{equation}
This equation contains 2 unknown parameters: initial position of the center of mass $\mathbf{r}_0$ and its velocity $\mathbf{v}_c$. These parameters can be (presumably) obtained by requiring that the equations of motion hold (since the law of interaction is known).
UPDATE:
To obtain $\mathbf{r}_0$ and $\mathbf{v}_c$ from the equations of motion:
Assume that the potential energy is: $U=\sum_{i=1}^{N}\sum_{k=i+1}^{N+1}U(|\mathbf{r}_i-\mathbf{r}_k|)$. Then the equation of motion for each particle is:
$$
m_i\mathbf{\ddot{r}}_i=-\sum_{k=1}^{N+1}U'(|\mathbf{r}_i-\mathbf{r}_k|)\frac{\mathbf{r}_i-\mathbf{r}_k}{|\mathbf{r}_i-\mathbf{r}_k|}
$$
For the first particle:
$$\tag{2}
m_1\mathbf{\ddot{r}}_1=-\sum_{k=1}^{N}U'(|\mathbf{r}_1-\mathbf{r}_k|)\frac{\mathbf{r}_1-\mathbf{r}_k}{|\mathbf{r}_1-\mathbf{r}_k|}-U'(|\mathbf{r}_1-\mathbf{r}_{N+1}|)\frac{\mathbf{r}_1-\mathbf{r}_{N+1}}{|\mathbf{r}_{1}-\mathbf{r}_{N+1}|}
$$
Substituting solution (1) into equation (2) and setting $t=0$ leads to an equation for $\mathbf{r}_0$. Of course, the equation can be nonlinear and it can have multiple solutions.
After $\mathbf{r}_0$ is found, $\mathbf{v}_c$ can be obtained from the same equation (2) for $t>0$.
