We are given,

a+bx

a−bx

=

b+cx

b−cx

=

c+dx

c−dx

(x ≠ 0)
Applying componendo and dividendo, we get

a+bx+a−bx

a+bx−a+bx

=

b+cx+b−cx

b+cx−b+cx

=

c+dx+c−dx

c+dx−c+dx

⇒

2a

2bx

=

2b

2cx

=

2c

2dx

⇒

a

b

=

b

c

=

c

d

Taking reciprocals, we get

b

a

=

c

b

=

d

c

Hence, a, b, c, d are in G.P.