## zmudz one year ago A sequence of real numbers (x_n) is defined recursively as follows: x_0=a and x_1=b are positive real numbers, and x_{n + 2} = (1 + x_{n + 1})/(x_n) for n = 0, 1, 2, .... so on. Find the value of x_{2012}, in terms of x_0 and x_1. Thanks!

1. danica518

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2. danica518

like this right?

3. danica518

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4. danica518

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5. danica518

im not seeing a trick, maybe its one of those continued fraction tricks?

6. danica518

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7. freckles

the sequence starts to repeat itself starting at x_5

8. danica518

really

9. freckles

yep

10. danica518

should i simplify X3 and X4 or can i leave it all in that form

11. freckles

simplifying helps to see the pattern

12. danica518

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13. freckles

$x_4=\frac{1+\frac{a+b+1}{ab}}{\frac{1+b}{a}} \\ \text{ multiply top and bottom by } ab \\ \\ x_4=\frac{ab +a+b+1}{b(1+b)}=\frac{a(b+1)+(b+1)}{b(b+1)} \\ x_4=\frac{(b+1)(a+1)}{b(b+1)}=\frac{a+1}{b}$

14. zmudz

That answer isn't right, and I'm not sure why because it makes sense to me... I don't know the answer but my homework is telling me it is wrong.

15. anonymous

the sequence is an example of one in rank-2 cluster algebras; it has a period of 5 for any initial values, so $$x_{2012}=x_{2012-2010}=x_2$$

16. anonymous

and we have $$x_2=\frac{1+x_1}{x_0}=\frac{1+b}a$$