# Finding a general formula for a sequence (x_k)

#### tommietang

##### New member
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

Staff member
Let's look at the first recursion:

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

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

#### tommietang

##### New member
Let's look at the first recursion:

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

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

Staff member
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:

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

This factors nicely to give integral roots...

#### tommietang

##### New member
No, the characteristic equation is:

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

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.