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

Solution:

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