Thread: Using induction to prove that every element in sequence is >= 0

I am working on an assignment in my calculus class. The first task is:

Given a recursively defined sequence:
a(1) = 2, a(n+1) = (a(n)^2 + 2) / 2a(n)

a) Use induction to prove that a(n) >= 0 for all n

I really don't know where to start. I have used induction to prove properties of sums, but when it comes to sequences, and especially recursive ones, I am blank.

Sorry for my potentially broken english!

Originally Posted by bjorno

I am working on an assignment in my calculus class. The first task is:

Given a recursively defined sequence:
a(1) = 2, a(n+1) = (a(n)^2 + 2) / 2a(n)

a) Use induction to prove that a(n) >= 0 for all n

I really don't know where to start. I have used induction to prove properties of sums, but when it comes to sequences, and especially recursive ones, I am blank.

Sorry for my potentially broken english!

Start by following the three steps that form proof by induction. Where are you stuck in doing this? Please show your attempt.

First i find a(2) = 6/4 = 3/2.
I'm already facing problems. I am used to work with sums (just + 1 on one side, and put in (n+1) on the other side, then rewrite until expressions look the same).
Don't know how to go about it from here.

Originally Posted by bjorno

First i find a(2) = 6/4 = 3/2.
I'm already facing problems. I am used to work with sums (just + 1 on one side, and put in (n+1) on the other side, then rewrite until expressions look the same).
Don't know how to go about it from here.

Is a(2) > 0? So, is Step 1 of the proof by induction true?

What is step 2? How does step 3 start? I suggest you think about each stpe of proof by induction more carefully so that you understand what each step is asking you to do.

I thought about it, and after a while, it just became clear to me! Thanks!