Mark the correct alternative in each of the following: If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is

Question

Philoid · Accepted Answer

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.

⇒