So A ∪ B = A + B − A ∩ B
A ∪ B ∪ C = A + B + C - (A ∩ B) - (A ∩ C) - (B ∩ C) + (A ∩ B ∩ C)
So i see where this is going...
Now to prove by induction how would i start?
start by assigning a natural number to something. i suggest calling your sets $\displaystyle \{A_j : j\in \{1,2,\dots, k\}\}$ and using induction on k.
that is, you are doing induction on "the number of sets you are taking the union of". your base case is k = 2, although you have also listed k=3.
one hopes you have previously established that (A U B) U C = A U (B U C), so that A U B U C is unambiguous.