(i) a2, b2, c2
Given that a, b, c are in GP.
By using the property of geometric mean,
b2 = ac
on squaring both the sides we get,
(b2)2 = (ac)2
(b2)2 = a2c2
∴ a2, b2, c2 are in G.P.
(ii) a3, b3, c3
Given that a, b, c are in GP.
By using the property of geometric mean,
b2 = ac
on squaring both the sides we get,
(b2)3 = (ac)3
(b2)3 = a3c3
(b3)2 = a3c3
∴ a3, b3, c3 are in G.P.
(iii) a2 + b2, ab + bc, b2 + c2
Given that a, b, c are in GP.
By using the property of geometric mean,
b2 = ac
a2 + b2, ab + bc, b2 + c2 or (ab + bc)2 = (a2 + b2) (b2 + c2) [by using the property of GM]
Let us consider LHS: (ab + bc)2
Upon expansion we get,
(ab + bc)2 = a2b2 + 2ab2c + b2c2
= a2b2 + 2b2(b2) + b2c2 [Since, ac = b2]
= a2b2 + 2b4 + b2c2
= a2b2 + b4 + a2c2 + b2c2 {again using b2 = ac }
= b2(b2 + a2) + c2(a2 + b2)
= (a2 + b2)(b2 + c2)
= RHS
∴ LHS = RHS
Hence a2 + b2, ab + bc, b2 + c2 are in GP.