let a and b be positive integer show that root 2 always liesbetween a upon b and a+2b upon a+b

Hello Keshav !

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 < b ...then a/b is always less than 1 ...and hence 1.414 will always fall right side of the a/b value...on real axis ...means greater than a/b .

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 < b ..

And i have shown above that ..how for a<b ..  a/b is less than 1.414 and (a+2b)/(a+b) is greater than 1.414

Cheers :)

regards

Yagya

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