Write the general equation for the Coulombian force on a charge ${q_1}$ due to a system of charges ${q_1}, {q_2}, \dots, {q_n}$.

According to the principle of superposition,the total force on a charge ${q_1}$ due to a system of $n$ point charges is the vector sum of the individual forces exerted by each charge on ${q_1}$.
Let $\vec{r_1}$ be the position vector of charge ${q_1}$ and $\vec{r_i}$ be the position vector of the $i$-th charge ${q_i}$.
The force exerted by charge ${q_i}$ on ${q_1}$ is given by Coulomb's law:
$\vec{F_{1i}} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_i}{|\vec{r_1} - \vec{r_i}|^3} (\vec{r_1} - \vec{r_i})$
The total force $\vec{F_1}$ on charge ${q_1}$ is the sum of these forces for $i = 2$ to $n$:
$\vec{F_1} = \sum_{i=2}^{n} \vec{F_{1i}} = \frac{q_1}{4\pi\epsilon_0} \sum_{i=2}^{n} \frac{q_i}{|\vec{r_1} - \vec{r_i}|^3} (\vec{r_1} - \vec{r_i})$