Question

If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is

A. p-q/q-r

B. q-r/p-q

C. pqr

D. none of these

Answer 1

Let the first term of AP be a and the common difference be d.

⇒ A_n = a+(n -1)d

p^th term is a+(p -1)d.

q^th term is a+(q -1)d.

r^th term is a+(r -1)d.

As they are in GP.

Let the first term be A and common ratio be R.

⇒ a+(p -1)d = A…(1)

⇒ a+(q -1)d = AR…(2)

⇒ a+(r -1)d = AR²…(3)

Subtracting 2 from 1 we get…

AR - A = a+(q -1)d - (a+(p -1)d)

= a+qd - d - a - pd+d

AR - A = qd - pd…

Subtracting 2 from 1 we get…

AR² - AR = a+(r -1)d - (a+(q -1)d)

= a+rd - d - a - qd+d

R(AR - A) = rd - qd.