Thread: sequences problem.

We have these two sequences, defined as:

Xn+1= Xn + Yn and Yn+1=Xn-Yn

and X0=Y0=1.

Find simple formulae for Xn and Yn. Don't even know how to start this... any help is welcome. I know it must be simple, but don't know how to do it. Usual difference equations I had were x smth and then I could use lambda exchange and find coefficients to solve it. This I don't know.

Originally Posted by fiksi

We have these two sequences, defined as:
Xn+1= Xn + Yn and Yn+1=Xn-Yn and X0=Y0=1.

Here are 16 terms of the sequence. See if you can find a pattern.

Originally Posted by Plato

Here are 16 terms of the sequence. See if you can find a pattern.

Well, then I'd have "guesswork" which wouldn't be rigorous enough for my teacher...
It looks as if I need a precise method here...

Originally Posted by fiksi

Well, then I'd have "guesswork" which wouldn't be rigorous enough for my teacher...It looks as if I need a precise method here...

Well OK! I will give you the y-sequence.
$\displaystyle y_n = \left( { - 1} \right)^n \cdot 2^{\left\lfloor {\frac{n}{2}} \right\rfloor } $ where $\displaystyle {\left\lfloor {\frac{n}{2}} \right\rfloor }$ is the floor function (greatest integer).

Now you try the x-sequence. Hint it involves the mod(2) function.