SOLUTION: The sequence u_n is defined recursively by the rules u_1 = 2, u_2 = 4, u_(n+2) = 4u_(n+1) + 5u_n for all n ∈ N. Prove the general term formula u

SOLUTION: The sequence u_n is defined recursively by the rules u_1 = 2, u_2 = 4, u_(n+2) = 4u_(n+1) + 5u_n for all n ∈ N. Prove the general term formula u_n = 5^n + (−1)^n.

Question 1033742: The sequence u_n is defined recursively by the rules
u_1 = 2, u_2 = 4, u_(n+2) = 4u_(n+1) + 5u_n for all n ∈ N.
Prove the general term formula u_n = 5^n + (−1)^n.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Just to make a CORRECTION: I believe the formula you want to be proved is

This is what we proceed to prove.
---------------------------------------------------------------------------
Proof by induction:
If n = 3, the recurrence formula gives .
Incidentally, the formula checks out.
Now assume that, up to a certain natural number k, is true.
We show that .
Now
==>
=
=
=
=, since .
Therefore, , and the theorem is proved by induction.
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SOLUTION: The sequence u_n is defined recursively by the rules u_1 = 2, u_2 = 4, u_(n+2) = 4u_(n+1) + 5u_n for all n &#8712; N. Prove the general term formula u_n = 5^n + (&#8722;1)^n.

SOLUTION: The sequence u_n is defined recursively by the rules u_1 = 2, u_2 = 4, u_(n+2) = 4u_(n+1) + 5u_n for all n ∈ N. Prove the general term formula u_n = 5^n + (−1)^n.