Question

let a and b be positive integer. show that √2 always lies between a/b and (a+2b)/(a+b)
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Answer 1

Answer:

Let us put condition on the second fraction, i.e., a+2ba+b . First, let us assume a+2ba+b>2–√2

a+2ba+b>2–√2

⟹a+2b>2–√2(a+b),

⟹2–√b(2–√−1)>a(2–√−1),

⟹2–√b>a,

⟹ab<2–√.

So, ab<2–√<a+2ba+b .

Similarly by assuming a+2ba+b<2–√ , we will get a+2ba+b<2–√<ab .

So, 2–√ lies either in the interval [ab,a+2ba+b] or [a+2ba+b,ab] for a>0 , b>0 and ab≠2–√.