Proving Positivity of a Recursively Defined Sequence using Induction

[nugget]

Homework Statement

A sequence x_n is defined recursively by x₁ = 2 and x_n+1 = 8x_n-1/4x_n+3 for n ∈ N.
a. Prove by induction or otherwise that x_n -1 > 0 for all n ∈ N.

Homework Equations

given...

The Attempt at a Solution

x₁ is given as 2, therefore x₁ - 1 = 1 > 0, which satisfies the criteria.

I'm not exactly sure how to employ induction (or the 'otherwise' method), from what I understand I need to prove that x_n+1 - 1 > 0, as then the base case (n=1) and all other cases are proved... on the right track?

 

[LCKurtz]

Is that middle term -(1/4)x_n or -1/(4x_n)?

 

[hgfalling]

Yes, you seem to be on the right track.

To use induction, you need to prove two things:

1) that your statement is true for a particular case.
2) that your statement being true for one case implies that it is true for the next case.

I assume you meant to write (8x_n - 1)/(4x_n + 3) there.

So 1) is easy, you already proved it.
For 2), can you show that if x_n > 1, x_n+1 > 1 for all n $$\in$$ N?