Proving convergence of Sequence dependent on previous terms

[tallandpoofy]

Homework Statement

Let x₁ > 9000, and

x_n+1 = )2009x_n + 2010)/2011 for n >1

show that (x_n) converges and find its limit

Homework Equations

Definition of a limit, Monotone Convergence Theorem.

The Attempt at a Solution

Since x_n+1 is monotone for n>1 and bounded, then it converges by Monotone Convergence Theorem.

How do i prove monotone for this function?
I tried x_n+2 - x_n+1 < 0 but it does not work since x_n+1 is dependent on x_n

 

[PAllen]

Homework Statement

Let x₁ > 9000, and

x_n+1 = )2009x_n + 2010)/2011 for n >1

show that (x_n) converges and find its limit

Homework Equations

Definition of a limit, Monotone Convergence Theorem.

The Attempt at a Solution

Since x_n+1 is monotone for n>1 and bounded, then it converges by Monotone Convergence Theorem.

How do i prove monotone for this function?
I tried x_n+2 - x_n+1 < 0 but it does not work since x_n+1 is dependent on x_n

You should be able to use your approach to prove monotonicity. Compute nth term minus (n+1)th term in terms of nth term (equivalent to your suggestion). Then analyze under what conditions it is positive (meaning sequence is monotonically decreasing). This analysis should also give you a clue about the limit.