Question

Let a and b be the positive integers. Show that ✓2 always lies between a/b and a+2b/a+b..
Chapter: Real numbers..​

Answer 1

Step-by-step explanation:

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

a+2ba+b>2–√,a+2ba+b>2,

⟹a+2b>2–√(a+b),⟹a+2b>2(a+b),

⟹2–√b(2–√−1)>a(2–√−1),⟹2b(2−1)>a(2−1),

⟹2–√b>a,⟹2b>a,

⟹ab<2–√.⟹ab<2.

So, ab<2–√<a+2ba+bab<2<a+2ba+b.

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

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