(N/A) The electronic configuration of $N$ atom $(Z=7)$ is $1s^2 2s^2 2p_x^1 2p_y^1 2p_z^1$. Total number of electrons in $N_2$ is $14$. The increasing order of energy levels for $N_2$ is: $\sigma 1s < \sigma^* 1s < \sigma 2s < \sigma^* 2s < \pi 2p_x = \pi 2p_y < \sigma 2p_z < \pi^* 2p_x = \pi^* 2p_y < \sigma^* 2p_z$.
$(i)$ $N_2$ ($14$ $e^-$): Configuration: $(\sigma 1s)^2, (\sigma^* 1s)^2, (\sigma 2s)^2, (\sigma^* 2s)^2, (\pi 2p_x)^2, (\pi 2p_y)^2, (\sigma 2p_z)^2$. Bond Order $(BO)$ = $\frac{1}{2}(10-4) = 3$. Diamagnetic (no unpaired electrons).
$(ii)$ $N_2^+$ ($13$ $e^-$): Configuration: $(\sigma 1s)^2, (\sigma^* 1s)^2, (\sigma 2s)^2, (\sigma^* 2s)^2, (\pi 2p_x)^2, (\pi 2p_y)^2, (\sigma 2p_z)^1$. $BO = \frac{1}{2}(9-4) = 2.5$. Paramagnetic (one unpaired electron).
$(iii)$ $N_2^-$ ($15$ $e^-$): Configuration: $(\sigma 1s)^2, (\sigma^* 1s)^2, (\sigma 2s)^2, (\sigma^* 2s)^2, (\pi 2p_x)^2, (\pi 2p_y)^2, (\sigma 2p_z)^2, (\pi^* 2p_x)^1$. $BO = \frac{1}{2}(10-5) = 2.5$. Paramagnetic (one unpaired electron).
$(iv)$ $N_2^{2+}$ ($12$ $e^-$): Configuration: $(\sigma 1s)^2, (\sigma^* 1s)^2, (\sigma 2s)^2, (\sigma^* 2s)^2, (\pi 2p_x)^2, (\pi 2p_y)^2$. $BO = \frac{1}{2}(8-4) = 2$. Diamagnetic (no unpaired electrons).
Stability order based on $BO$: $N_2 > N_2^+ \approx N_2^- > N_2^{2+}$.