To prove: Prove that a:b = (2+ √3) : (2 - √3)
Given: Arithmetic mean is twice of Geometric mean.
Formula used: (i) Arithmetic mean between a and b = \(\frac{a+ b}{2}\)
(ii) Geometric mean between a and b = \(\sqrt{ab}\)
Am = 2(Gm)
\(\frac{a+ b}{2}\) = 2 (\(\sqrt{ab}\))
⇒ a + b = 4\(\sqrt{ab}\)
Squaring both side
⇒ (a + b)2 = 16ab … (i)
We know that (a – b)2 = (a + b)2 – 4ab
From eqn. (i)
⇒ (a – b)2 = 16ab – 4ab
⇒ (a – b)2 = 12ab … (ii)
Dividing eqn. (i) and (ii)
\(\frac{(a + b)^2}{(a-b)^2} = \frac{16ab}{12ab}\)
\(\left(\frac{a+b}{a-b}\right)^2\) = \(\frac{16}{12}\)
Taking square root both side
Applying componendo and dividend
Hence Proved