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!

 

[stapel]

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

What methods or formulas have they given you in your class? What have you tried so far? Where are you stuck?

Please pick one of the three exercises and reply with your efforts thus far on that one, so we can see what you're doing and where you're having difficulty. Thank you!

 

[pka]

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

\(\displaystyle \begin{array}{l} x_2=x_{0+2}=-x_1+6x_0=~?\\x_3=-x_2+6x_1=~?\end{array} \)

The idea is to keep "building" the sequence until you see the pattern.

 

[tommietang]

What methods or formulas have they given you in your class? What have you tried so far? Where are you stuck?

Please pick one of the three exercises and reply with your efforts thus far on that one, so we can see what you're doing and where you're having difficulty. Thank you!

I've actually been sick recently and couldn't attend the lectures, so I am trying to figure out everything online.

I solved 1) now, by finding characteristic equation and the roots and solving with a system of equations. 2) is the same but just 1 more variable to solve. 3) just with an 'i', ezpz!