zmudz
  • zmudz
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!
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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danica518
  • danica518
|dw:1437194441049:dw|
danica518
  • danica518
like this right?
danica518
  • danica518
|dw:1437194518228:dw|

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danica518
  • danica518
|dw:1437194651461:dw|
danica518
  • danica518
im not seeing a trick, maybe its one of those continued fraction tricks?
danica518
  • danica518
|dw:1437194858049:dw|
freckles
  • freckles
the sequence starts to repeat itself starting at x_5
danica518
  • danica518
really
freckles
  • freckles
yep
danica518
  • danica518
should i simplify X3 and X4 or can i leave it all in that form
freckles
  • freckles
simplifying helps to see the pattern
danica518
  • danica518
|dw:1437195330961:dw|
freckles
  • freckles
\[x_4=\frac{1+\frac{a+b+1}{ab}}{\frac{1+b}{a}} \\ \text{ multiply top and bottom by } ab \\ \\ x_4=\frac{ab +a+b+1}{b(1+b)}=\frac{a(b+1)+(b+1)}{b(b+1)} \\ x_4=\frac{(b+1)(a+1)}{b(b+1)}=\frac{a+1}{b}\]
zmudz
  • 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.
anonymous
  • anonymous
the sequence is an example of one in rank-2 cluster algebras; it has a period of 5 for any initial values, so $$x_{2012}=x_{2012-2010}=x_2$$
anonymous
  • anonymous
and we have $$x_2=\frac{1+x_1}{x_0}=\frac{1+b}a$$

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