If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a – b, d – c are in G.P.

Question

Philoid · Accepted Answer

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)

⇒ (a – b)² = a(d – c)

As a, (a – b), (d – c) satisfy the geometric mean relationship

Hence a, (a – b),(d – c) are in G.P.