MHB Finding a general formula for a sequence (x_k)

[tommietang]

I'm trying to do 3 questions, each one a bit more complex than the previous, but all have the same ideas. ( 2) has 1 more term than 1, 3) is with imaginary numbers)

Could someone please guide me on how to do them? Am I trying to substitute things into each other?

Suppose that the sequence x0, x1, x2... is defined by

1) x_0 = 4, x_1=1, x_(k+2) = -x_(k+1) + 6x_k

2) x_0 = 7, x_1=4, x_2=7, x_(k+3) = -5x_(k+2) + 2x_(k+1) + 24x_k

3) x_0 = 3, x_1=1, x_(k+2) = -6x_(k+1) - 10x_k

All 3 for k>=0
Find a general formula for x_k

I would greatly appreciate any help!

 

[MarkFL]

Let's look at the first recursion:

$$x_{k+2}=-x_{k+1}+6x_{k}$$

Can you state the characteristic equation and then find its roots?

 

[tommietang]

Let's look at the first recursion:

$$x_{k+2}=-x_{k+1}+6x_{k}$$

Can you state the characteristic equation and then find its roots?

Is it x^2 = -x + 1
Solving for root: x = 1/2(-1 - sqrt(5)) and 1/2(sqrt(5)-1)

 

[MarkFL]

Is it x^2 = -x + 1
Solving for root: x = 1/2(-1 - sqrt(5)) and 1/2(sqrt(5)-1)

No, the characteristic equation is:

$$r^2+r-6=0$$

This factors nicely to give integral roots...

 

[tommietang]

No, the characteristic equation is:

$$r^2+r-6=0$$

This factors nicely to give integral roots...

Thank you sir, I wasn't sure how to approach the problem because I couldn't attend recent lectures. But this tip gave me the ability to solve all 3.

 

[chisigma]

Thank you sir, I wasn't sure how to approach the problem because I couldn't attend recent lectures. But this tip gave me the ability to solve all 3.

All the difference equation are linear homogeneous with constant coefficients and the solving procedure is illustrated here...

http://mathhelpboards.com/discrete-mathematics-set-theory-logic-15/difference-equation-tutorial-draft-part-ii-860.html

Kind regards

$\chi$ $\sigma$