Question

If a,b,c are in A.P .prove that (a-c)2 = 4 ( b2 - ac).

Answer 1

Since a, b, c are in A.P so

                       b – a = c – b

                      b + b = c + a

                         2 b = c + a

  Taking squares on both sides

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

                      4b² = (c + a)²

Subtracting by 4ac on both sides, we get

                4b² – 4ac  = (c + a)² – 4 ac

                4(b² – ac)  = c² + a² + 2 ac – 4 ac

                4(b² – ac)  = c² + a² - 2 ac

                4(b² – ac)  = ( c – a )²