Ok at first glance this seemed pretty easy, but I'm having a lot trouble with it. Let a,b be positive integers. Show that always lies between the two fractions a/b and (a+2b)/(a+b). Which fraction is closer to . Any help or hint would be appreciated.
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Oh alright, I need to learn how to manipulate surds better...
The case where just requires the inequality signs switched.
Now, recover the first case where
Suppose if possible that
Then , a contradiction. Then
A similar argument can be applied to the other case.
Is there a direct way to show this? I find using contradiction to be a little like cheating in this problem . The reason I didn't was that I wasn't sure my inequalities were valid when trying to recover the absolute value sign.
For the last part, can't we consider the differences between and , and and ?
Which are:andClearly the second of these expressions is less than the first. So is closer to . (Incidentally, doesn't the fact that and have opposite signs prove the first part of the question as well?)