If a, b, c are in A.P. b, c, d are in G.P. and are in A.P., prove that a, c, e are in G.P.
Given:
a,b,c are in AP
∴ 2b = a + c …… (i)
b,c,d are in GP;
⇒ c2 = bd …… (ii)
1/c, 1/d, 1/e are in AP;
⇒
⇒ …(iii)
From the above substituting for b & d in (ii) above,
⇒
⇒ c(c + e) = (a + c) e
⇒ c2 + ce = ae + ce
⇒ c2 = ae
Thus a, c, e are in GP