SOLUTION: The sequence u_n is defined recursively by the rules u_1 = 3, u_2 = 3, u_(n+2) = u_(n+1) + 2u_n for all n ∈ N. Find the formula for the general term of the sequence and pro

Algebra ->  Proofs -> SOLUTION: The sequence u_n is defined recursively by the rules u_1 = 3, u_2 = 3, u_(n+2) = u_(n+1) + 2u_n for all n ∈ N. Find the formula for the general term of the sequence and pro      Log On


   

Question 1021209: The sequence u_n is defined recursively by the rules
u_1 = 3, u_2 = 3, u_(n+2) = u_(n+1) + 2u_n for all n ∈ N.
Find the formula for the general term of the sequence and prove it.
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
The characteristic polynomial is , which has roots . So the general term is of the form for some constants a and b.

When n = 1, we have .

When n = 2, we have

Solving for a and b, we get a = 1, b = -1. So the general formula is

for all .