Q21 of 176 Page 20

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.

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.

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 three distinct real numbers in G.P. and a + b + c = xb, then prove that either x < – 1 or x > 3.

If p^th, q^th and r^th terms of an A.P. and G.P. are both a, b and c respectively, show that a^{b – c}b^c ^{– a} c^{a – b} = 1.