How do you calculate the delta H for multiple reactions?
Find the energy interval that produces atomic nitrogen in the flask but not any ions


This is the key. The question wants you to determine how much energy you need to supply just to only have atomic nitrogen, no ions.

If you supply $941~kJ/mol$, then you have supplied enough energy to move reaction 1 forward. In your flask, you would have $N(g)$ inside. Now, if you further supply energy up to $1402~kJ/mol$, then you have enough energy to make reaction 3 proceed, and that will convert your atomic nitrogen to nitrogen ions, so that's a no-no.

So you have to maintain $E$ that is in between $941~kJ/mol$ to $1402~kJ/mol$ to prevent reaction 3 from proceeding.

If you supply $1402~kJ/mol$ after breaking the bond, then you will definitely produce $N^+$ ions as you have met reaction 3's $E$ requirement.