Solution of the reccurrence relation

[romo84]

Homework Statement

Find the solution of the recurrence relation a_n=3a_n-1 - 3a_n-2 + a_n-3 if a₀=2, a₁=2, and a₂= 8.

Homework Equations

The Attempt at a Solution

I know that I need to express a_n in terms of a_n-1 but don't know how to conclude.

 

[story645]

I know that I need to express a_n in terms of a_n-1 but don't know how to conclude.

Um a_n is already in terms of a_n-1 so what do you mean? What are you covering in class now that might make the solution clearer? Usually solving for a recurrence relation means finding the general (non-recursive) form. Is that what you're supposed to do here?

 

[romo84]

Yes, I am to find the non-recurrsive form (closed form).

Thank You!

 

[stevenb]

These types of equations are solvable using the Z-transform which is the dicrete math version of the Laplace transform. Are you familiar with this approach?

The basic idea is to take the Z-transform of your difference equation, then solve, then take the inverse Z-transform.

 

[romo84]

It is actually to be solved using "Linear homogeneous recurrence relations" using the characteristic equation. and a2 = 4, not 8.

 

[stevenb]

It is actually to be solved using "Linear homogeneous recurrence relations" using the characteristic equation. and a2 = 4, not 8.

Oh, too bad they make you do it the hard way. A couple of turns of the Z-crank yield a_n=n²-n+2.

Anyway, having the answer will help you check that you have your method right.

 

[vela]

It is actually to be solved using "Linear homogeneous recurrence relations" using the characteristic equation. and a2 = 4, not 8.

What did you get for the characteristic equation and its roots?