If a, b, c are in continued proportion, then prove that

(i)

(ii)

(i) Given: a, b, c are in continued proportion.

∴ b² = ac

If then:

a(a - 4c) = (a – 2b)(a + 2b)

⇒ a² – 4ac = a² – 4b²

⇒ -4ac = -4b²

⇒ b² = ac, which holds true as the numbers are in continued proportion.

∴

(ii) Given: a, b, c are in continued proportion.

∴ b² = ac

If then:

b(a - c) = (a – b)(b + c)

⇒ ab – bc = ab + ac – b² - bc

⇒ ab – bc - ab - ac + bc = -b²

⇒ -ac = -b²

⇒ b² = ac, which holds true as the numbers are in continued proportion.

∴