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

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

Grade:10

1 Answers

Arun
25758 Points
4 years ago
 

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 :)

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