Question

If pth, qth and rth terms of A. P A1, A2, A3,.... are in G. p then ap:aq​

Answer 1

Answer:

In AP, first term be a

Difference be d

∴p

th

term ⇒a+(p−1)d

q

th

term a+(q−1)d

r

th

term a+(r−1)d

since they are in GP

and let first term in G.P be A

and common ration be r

∴ As given

a+(p−1)d=A...(1)

a+(q−1)d=Ar...(2)

a+(r−1d)=Ar

2

...(3)

subtracting (1) from (2)

[(q−1)−(p−1)]d=A[r−1]

(q−p)d=A(r−1)...(4)

subtracting (2) form (3)

(r−q)d=Ar(r−1)...(5)

Dividing (5), we get

A(r−1)

Ar(r−1)

​

=

(q−p)d

(r−q)d

​

∴r=

p−q

q−r

​

Hence proved

Step-by-step explanation: