Question

If a, b, c are in continued proportion and a(b - c) = 2b, prove that `a - c = (2(a + b))/a`

Answer 1

Since a, b, c are in continued proportion,

`a/b = b/c`

⇒ b² = ac

∴ a(b - c) = 2b

⇒ ab - ac = 2b

⇒ ab - b² = 2b

⇒ b(a - b) = 2b

⇒ a - b = 2

Now

L.H.S = a - c

`= (a(a - c))/a`

`= (a^2 - ac)/a`

`= (a^2 -b^2)/a`

`= ((a - b)(a + b))/a`

`= (2(a + b))/a`

= R.H.S.