SOLUTION: hi am stuck on a question on theory of numbers show that, if p/q is a good approximation of &#8730;2 then (p�+2q�)/(2pq) is a better one. starting with p=q=1, show that &#8730;2

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Question 461622: hi am stuck on a question on theory of numbers
show that, if p/q is a good approximation of √2 then (p�+2q�)/(2pq) is a better one. starting with p=q=1, show that √2 ~ 577/408 and estimate the accuracy of the approximation.
Answer by robertb(5830) (Show Source):
You can put this solution on YOUR website!
I will only answer the first part of the problem. I leave the iteration part to you.
Let p, q be positive numbers. Without loss of generality, we can let . I will show that . (If the arguments are similar.)
(I) ==>
==> because is 1-to-1 over the positive numbers.
==>
==>
<==>
(II) By the AM-GM inequality,

<==>
==>
Then combining the results of (I) and (II), we get , and the proof is complete.

SOLUTION: hi am stuck on a question on theory of numbers show that, if p/q is a good approximation of &#8730;2 then (p�+2q�)/(2pq) is a better one. starting with p=q=1, show that &#8730;2

SOLUTION: hi am stuck on a question on theory of numbers show that, if p/q is a good approximation of √2 then (p�+2q�)/(2pq) is a better one. starting with p=q=1, show that √2