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