This is the full question: Find all pure strategy Nash equilibria of the Cournot oligopoly game with N firms facing linear demand $P = a − Q$ when the total cost for each firm is $c_i(q_i) = q_i^2 $
My understanding is that:
$$\Pi_i =q_i(a − Q) - q_i^2$$
$$\max_{q_i} [qi(a − Q) - qi^2]$$
Therefore, the FOC is:
$$a-Q-2q_i=0$$
This is the FOC for each firm.
This is the part that I get stuck at. How do you add all the FOCs of N firms together? Thank you!