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It is my understanding that if we have a sequence defined as follows:

a_{n+1}=(ψ)a_{n}+ (λ)

Then if ψ≥1 or ψ≤-1, the sequence diverges. If -1<ψ<1, the sequence converges to:

λ/(1-ψ)

I was working problems in a book and one of the problems said that the following sequence converges:

x_{n}=√(5x_{n-1}+6), where x_{1}=2

Can somebody explain to me why this converges? I mean, I guess it grows at such a slow rate that eventually it converges to a set limit. But what are some ways I can determine that limit? I mean are there any tools to evaluate the convergence of recurrance relations, like basic tools we have when dealing with regular sequences? The only "tool" I know is the one I listed above and my calculus book doesn't really look at these. Or when it comes to these, do I just have to plug in the first few numbers and look for a general trend? Thanks for the advice!

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# Recurrence Relation - limit of a sequence

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