Question

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

a-c = 2(a + b) / a

Answer 1

Given a , b , c are in continued proportion
⇒ a : b : : b : c ⇒ a x c = b x b
ac=b² .............(1)

now given that a(b-c)=2b ⇒ ab - ac -2b = 0
⇒ ab - b² -2b = 0
⇒ b(a - b -2) = 0
⇒ a - b -2 =0 here b≠0
⇒ a - b = 2 ............(2)

now we have to prove that a-c=2(a+b)/a
LHS = a-c ⇒ multipying a to nr. and dr.

⇒ a(a-c)/a ⇒ (a² - ac)/ a ⇒ (a² - b²) / a here put ac=b² from equation (1)
⇒ (a-b)(a+b)/a ⇒ 2(a+b)/a from equations (1) and (2)
so a- c = 2(a+b)/a