Solving Linear Recurrence: ax_k+1 + bx_k + c

[Parth Dave]

Consider the recurrence x_k+2 = ax_k+1 + bx_k + c where c may not be zero.

If a + b is not equal to 1 show that p can be found such that, if we set y_k = x_k + p, then y_k+2 = ay_k+1 + by_k. [Hence, the sequence x_k can be found provided y_k can be found]


First of all, sorry about the messiness, I don't know how to use LaTeX. Now, this is the question exactly as it is from the question sheet. My problem is, I don't understand the question. And its kind of really hard to start the question without understanding it . My biggest concern is, what the heck is p and where does it come from? The way I read it, p is just -c.

Thx in advance for any help.

 

[dextercioby]

1.You should learn "tex".
2.Hypothetis:$$x_{k+2}=ax_{k+1}+bx_{k}+c$$ (1)
$$y_{k}=x_{k}+p$$ (2)
$$y_{k+2}=ay_{k+1}+by_{k}$$ (3)
3.Question:$$p=...?$$

4.From (2) u have:
$$y_{k+2}=x_{k+2}+p$$ (4)
Combining (1) and (4),u get:
$$y_{k+2}=ax_{k+1}+bx_{k}+c+p$$ (5)
Equate (5) with (3),make use of (2) and extract 'p':

Answer:$$p=\frac{c}{a+b-1}$$

Daniel.

 

[Parth Dave]

Ah, it all makes sense. Can't believe I never saw that. Thx a lot!