mkwok said:
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 < y
0 < x
0, 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____y
0____________________________x
0
Now, we have: x
1 = (x
0 + y
0) / 2, that means, x
1 lies exactly at
the middle of x
0, and y
0.
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 x
1 > y
1.
0____y
0____
y1_________
x1_______________x
0
Now, where do x
2, and y
2 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? :)
