Q19 of 176 Page 20

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

⇒ c^{2 =} ae

Thus a, c, e are in GP

If are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.

If x^a = x^b/2z^b/2 = z^c, then prove that are in A.P.

If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x², b², y² are in A.P.

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.