Question

Let a and b be the positive integers.show that root2 always lies between a divided by b and a plus 2b divided by a plus b

Answer 1

Answer:

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

a+2ba+b>2–√,

⟹a+2b>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–√.