If a, b, c are in A.P., prove that:
(a - c)2 = 4 (a - b) (b - c)
(a - c)2 = 4 (a - b) (b - c)
a2 + c2 - 2ac = 4(ab – ac – b2 + bc)
a2 + 4c2b2 + 2ac - 4ab - 4bc = 0
(a + c - 2b)2 = 0
a + c - 2b = 0
Since a, b, c are in AP
b - a = c - b
a + c - 2b = 0
Hence,
(a - c)2 = 4 (a - b) (b - c)