# Sequence and Series, finding relationship

1. Nov 25, 2007

### mkwok

1. The problem statement, all variables and given/known data

0<y0<x0
x1=(x0+y0)/2
y1=$$\sqrt{x0y0}$$

in general
Xn+1=(xn+yn)/2
Yn+1=$$\sqrt{XnYn}$$

2. Relevant equations

none

3. The attempt at a solution

I have no idea
I tried to solve for Xn and substituting that into another equation...
however, I don't know how to simplify it down to one single variable....

How do I take the limit of the sequence, so that I can find the relationship

2. Nov 25, 2007

### EnumaElish

Please state the problem as it is given.

3. Nov 25, 2007

### mkwok

the problem was given that way, we are trying to find the relationship between X and y

4. Nov 25, 2007

### rock.freak667

take the general equations and sub one into the other is all i can suggest

5. Nov 26, 2007

### HallsofIvy

Staff Emeritus
Since there was NO X or Y in what you gave, I don't see how that can be true.

6. Nov 26, 2007

### VietDao29

Hmm, I think I got what you mean.

Given 0 < y0 < x0, the sequence (xn), and (yn) are defined as follow:

$$x_{n + 1} = \frac{x_n + y_n}{2}, n \in \mathbb{N}$$ (1)
$$y_{n + 1} = \sqrt{x_n y_n}, n \in \mathbb{N}$$ (2)

Now find the limit of xn, and yn (Or does it tell you to prove that the limit of the two sequences are the same?).

Is that the correct problem?

---------------------------

Ok, now you must at least have a vision of how xn, and yn behave. Draw a pictures like this:

0____y0____________________________x0

Now, we have: x1 = (x0 + y0) / 2, that means, x1 lies exactly at the middle of x0, and y0.

$$y_1 = \sqrt{x_0 y_0} > \sqrt{y_0 ^ 2} = y_0$$

You'll also have: $$x_1 - y_1 = \frac{x_0 - 2\sqrt{x_0 y_0} + y_0}{2} = \frac{(\sqrt{x_0} - \sqrt{y_0}) ^ 2}{2} > 0$$, so x1 > y1.

0____y0____y1_________x1_______________x0

Now, where do x2, and y2 lie?

---------------------------

So, in conclusion, let's answer some questions:

1. xn, and yn, which is greater?

2. Is (xn) an increase sequence, or a decrease sequence? Can you prove it?

3. Is (yn) an increase sequence, or a decrease sequence? Can you prove it?

4. Are they bounded? Do they have limit?

5. Do they both have the same limit?

Ok, I think you can take it from here. Can you? :)

Last edited: Nov 26, 2007