Find all solutions of the recurrence relation
I know how to do the linear ones but if someone could point me in the right direction to complete this one, that would be great. Thanks!
EDIT: I know the answer is I just don't know how to get there.
Find all solutions of the recurrence relation
I know how to do the linear ones but if someone could point me in the right direction to complete this one, that would be great. Thanks!
EDIT: I know the answer is I just don't know how to get there.
This solution does not satisfy the recurrence equation. Indeed, according to this solution, Substituting this into the equation, we get , while it should be .I know the answer is
To solve the equation, follow these steps.
1. Note that when is fixed, the sequence is uniquely determined. Therefore, everything further will use some fixed parameter .
2. Write , , , and explicitly (using ). Hint: don't compute and don't add the powers of 2; for example, leave instead of 12 or .
3. Guess the regularity and write the general formula for . You may use the formula , so .
4. Prove by induction that this is indeed the solution. Namely, verify that is given by this formula, and that for every , if is given by the formula, then so is .
You could also use the Z transform to solve it. The idea there is very much along the same lines as the Laplace Transform.