Solution of the reccurrence relation

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Discussion Overview

The discussion revolves around solving the recurrence relation an=3an-1 - 3an-2 + an-3 with initial conditions a0=2, a1=2, and a2=8. Participants explore methods for finding a closed form solution, including the Z-transform and characteristic equations.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses uncertainty about how to express an in terms of an-1 and seeks guidance on concluding the solution.
  • Another participant clarifies that an is already in terms of an-1 and questions what concepts are currently being covered in class that might aid in finding the solution.
  • A participant confirms the goal is to find the non-recursive (closed) form of the relation.
  • One suggestion involves using the Z-transform, noting it is akin to the Laplace transform and proposing to take the Z-transform of the difference equation.
  • Another participant insists that the problem should be solved using "Linear homogeneous recurrence relations" and mentions a correction regarding the initial condition a2, stating it should be 4 instead of 8.
  • A follow-up comment suggests that using the Z-transform could yield a solution of an=n2-n+2, implying a simpler method exists.
  • A participant asks for the characteristic equation and its roots, indicating a desire to explore that method further.

Areas of Agreement / Disagreement

There is no consensus on the correct method to solve the recurrence relation, as participants propose different approaches and express differing views on the initial conditions.

Contextual Notes

Participants have not fully resolved the initial conditions, particularly regarding the value of a2, and there are multiple proposed methods for solving the recurrence relation, each with its own assumptions and implications.

romo84
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Homework Statement


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


Homework Equations





The Attempt at a Solution


I know that I need to express an in terms of an-1 but don't know how to conclude.
 
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romo84 said:
I know that I need to express an in terms of an-1 but don't know how to conclude.


Um an is already in terms of an-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?
 
Yes, I am to find the non-recurrsive form (closed form).

Thank You!
 
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.
 
Last edited:
It is actually to be solved using "Linear homogeneous recurrence relations" using the characteristic equation. and a2 = 4, not 8.
 
Last edited:
romo84 said:
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 an=n2-n+2.

Anyway, having the answer will help you check that you have your method right.
 
Last edited:
romo84 said:
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?
 

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