If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a – b), (d – c) are in G.P.
a, b, c are in AP
So, 2b = a + c …(1)
b, c, d are in GP
So, b2 = ad …(2)
Multiply first equation with a and subtract it from 2nd.
b2 – 2ab = ad – ac – a2
⇒ a2 + b2 – 2ab = a(d – c)
Hence a, (a – b), (d – c) are in G.P.