Let a and b be positive integers. Show that root2 always lies between a/b and a+2b/a+b

This one is  a bit tricky question ..

Value of root(2) = 1.414 

since a is an integer and b is also an positive integer so ..

a/b is a rational number ..! 

If a >b  then  a/b > 1

also value of a/b can be greater than 1.414 in some cases ...Like (2/1) ( so a>b can not be the option ...a must be less than b)

So condition is if a

Now   (a+2b)/(a+b)  can be written as    1 + b/(a+b) 

So it is always greater than one ... Critically when a=b ..it will be equal to 1.5 which is greater than 1.414

But once a > b .. the value can fall less than 1.414 ..example 2 and 1 ...value will be  1.333

So Root(2) always lies between  a/b and (a+2b)/(a+b)  only if  a  ..

And i have shown above that ..how for a

Cheers :)