# Thread: Linear second order recurrence sequence

1. ## Linear second order recurrence sequence

Can somebody please point me into a really helpful website / video to help me get my head around these, I under stand the basics but getting confused when you start adding

(fn2+3) + (fn)2 and (fn2+3) + fn2 N = 1,2,... N-3 etc

thanks, Matt

2. Sorry, I can't understand your notation

3. Note that if you go into the Advanced Editing mode, you can use the TeX button to enclose math tags around LaTeX code. Also note that subscripts can be written as a_{n}, which displays as $\displaystyle a_{n}.$

As Chris11 pointed out, it's unclear exactly what values are in the "subscript", or function argument. You also have n's and N's. Mathematics is case-sensitive, so you should standardize on one or the other. Typically, capital N's are used to denote a fixed integer value, whereas little n's are more used to denote a variable integer value.

4. What i'm looking for is help on understanding this and the cassini identity and how i can get from a fibonnaci expression such as;

$\displaystyle Un+1Un+2 - Un+1(Un)-Un(Un+2)$

using the cassini identity;

$\displaystyle UnUn+2 = U2n+1 + (-1)^n+1$

to show that

$\displaystyle Un+1Un+2-UnUn+3 = (-1)^n$

where;

$\displaystyle Un+1Un+2 - Un+1(Un)-Un(Un+2) = Un+1Un+2-UnUn+3 = (-1)^n$

5. I'm going to take a stab at re-writing your whole post, using my guesses for what you mean (you still need a lot of cleaning-up of notation). Please correct any errors.

What i'm looking for is help on understanding this and the cassini identity and how i can get from a fibonnaci expression such as;

$\displaystyle U_{n+1}U_{n+2} - U_{n+1}U_{n}-U_{n}U_{n+2}$

using the cassini identity;

$\displaystyle U_{n}U_{n+2} = U_{2n+1} + (-1)^{n+1}$

to show that

$\displaystyle U_{n+1}U_{n+2}-U_{n}U_{n+3} = (-1)^{n}$

where;

$\displaystyle U_{n+1}U_{n+2} - U_{n+1}U_{n}-U_{n}U_{n+2}=U_{n+1}U_{n+2}-U_{n}U_{n+3} = (-1)^{n}$.

How is that for the mathematical notation? As for your English, that needs cleaning up as well, since you have very convoluted syntax, if it's even a sentence at all.

6. Actbeet, yes thats what i meant, I'm just not as familiar to TEX as you might be, but you were able to understand what I had written, thanks.

7. Let's see if we (or should I say I) get this right.

Given $\displaystyle U_n +U_{n+1} = U_{n+2}$ (the Fibinacci sequence recursion relation)

and Cassini's identity

$\displaystyle U_{n-1}U_{n+1} - U_n^2 = (-1)^n$

prove

$\displaystyle U_{n+1}U_{n+2} - U_n U_{n+3} = (-1)^n$

Is this it?

8. Reply to Danny.

Perhaps, but it looks more to me like this:

Assume

$\displaystyle U_{n}U_{n+2}=U_{2n+1}+(-1)^{n+1}\quad(\text{Cassini}),$ and

$\displaystyle U_{n+1}U_{n+2}-U_{n+1}U_{n}-U_{n}U_{n+2}=(-1)^{n}.$ Show that

$\displaystyle U_{n+1}U_{n+2}-U_{n}U_{n+3}=(-1)^{n}.$

The fact that post # 4 uses the word "Fibonacci" may or may not imply that the OP means $\displaystyle U_{n}$ to be the Fibonacci sequence. He never actually posts that relation. That's my impression, but I could be wrong.

9. When I wikipedia'd Cassini it gave what I gave in post #7. I think we need the OP to give some clarification before anyone wastes their time working on an answer to what we "think they meant."

Just my two cents.

10. Originally Posted by Danny
When I wikipedia'd Cassini it gave what I gave in post #8. I think we need the OP to give some clarification before anyone wastes their time working on an answer to what we "think they meant."

Just my two cents.
Agreed.