# Thread: second order recurrence sequences

1. ## second order recurrence sequences

Hello,

I hope I have posted this in the correct place, if not, sorry.

Okay, I have just started working through the material, but I am having some problems working out one of the examples, which is given below:

Un+2 = 12Un+1 - 20Un (n = 0,1,2...)

We are given
U0 = 1, U1 = 2

and are asked to work out U2 to U4.

But, maybe my head is just blocked, but I can't seem to get the figures. I have looked at the answers at the back of the book, but still no luck.

So, if anyone could point me in the right direction, I would be very thankful.

Cheers
Sean

PS - Sorry about typing the question out, I haven't got the hang of LaTeX just yet.

2. Originally Posted by feely
Hello,

I hope I have posted this in the correct place, if not, sorry.

Okay, I have just started working through the material, but I am having some problems working out one of the examples, which is given below:

Un+2 = 12Un+1 - 20Un (n = 0,1,2...)

We are given
U0 = 1, U1 = 2

and are asked to work out U2 to U4.

But, maybe my head is just blocked, but I can't seem to get the figures. I have looked at the answers at the back of the book, but still no luck.

So, if anyone could point me in the right direction, I would be very thankful.

Cheers
Sean

PS - Sorry about typing the question out, I haven't got the hang of LaTeX just yet.
$U_{n+2} = 12U_{n+1} - 20U_n$

$U_{n + 2} - 12U_{n + 1} + 20U_n = 0$

Solve the characteristic equation
$m^2 - 12m + 20 = 0 \implies m = 2, 10$

So the general solution is going to be
$U_n = A \cdot 2^n + B \cdot 10^n$

$U_0 = 1 \implies A + B = 1$
$U_1 = 2 \implies 2A + 10B = 2$
$A = 1, ~B = 0$
$U_n = 2^n$