This is a so-called linear homogeneous recurrence relations with constant coefficients. See further down the same page how to solve it.I'm trying figure out a formula that would work with this sequence defined by a0=1, a1=2, and ak = 2ak-1 + 3ak-2, for all integers k is equal or greater than 2.

You can see this from a Venn diagram. Or you follow the usual method: proving and . For each inclusion, assume that some x is in the smaller set; then you need to show that x is in the bigger set.If A and B are subsets of the universal set U, then (AUB)^c = (A^c)⋂(B^c).

Edit.

No, one proves by induction facts about objects that are inductively generated. For example, natural numbers are generated from 0 by repeatedly applying "plus one" function. Sets in general are not generated gradually in this way, so one can't use induction here.or would i need to prove this by some form of induction?....