Question

1.If a,b and c are in ap ,show that bc-a2 ,ca-b2 and ab-c2 are in A.P.

Answer 1

Heya..!!!!!


Given that:-

a,b,c are in A.P

D = b-a = c-b

=> 2b = a+c-----------(1)

In A.P

D = (ca - b²) - (bc - a²) = (ab - c²) - (ca - b²)

=> (ab - c²) + (bc - a²) = 2(ca - b²)

=> (ab - c² + bc - a²)

=> b(a + c) - (a² + c²)

=> b(2b) - [ a² + c² + 2ac - 2ac ]---from--(1)

=> 2b² - [ (a + c)² - 2ac ]

=> 2b² - [ (2b)² - 2ac] ----from--(1)

=> 2b² - 4b² + 2ac

=> -2b² + 2ac

=> 2(ac - b²)

R.H.S = L.H.S