Suppose that b₀, b₁, b₂, . . . is a sequence defined as follows:

b₀ =1, b₁ =2, b₂ =3, and b_k = b_k1 + 4*b_k2 + 5*b_k3 for all integers k 3.

Prove by mathematical induction that b_n 3ⁿ for all integers n 0.