Sequence and Series, finding relationship

[mkwok]

Homework Statement

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

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

Homework Equations

none

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

 

[EnumaElish]

Please state the problem as it is given.

 

[mkwok]

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

 

[rock.freak667]

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

 

[HallsofIvy]

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

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

 

[VietDao29]

Homework Statement

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

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

Homework Equations

none

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

Hmm, I think I got what you mean.

Given 0 < y₀ < x₀, the sequence (x_n), and (y_n) 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 x_n, and y_n (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 x_n, and y_n 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. x_n, and y_n, which is greater?

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

3. Is (y_n) 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? :)