Ifp.9.r are in AP, show that the pth, qth and rth terms of any GP are in
GP.
Answers
Answered by
13
Answer:
Given,
pth,qth,rth,sth terms are in AP
a
p
=a+(p−1)d
a
q
=a+(q−1)d
a
r
=a+(r−1)d
a
s
=a+(s−1)d
a
p
,a
q
,a
r
,a
s
are in G.P.
a
p
a
q
=
a
q
a
r
=
a
r
a
s
----------------------------------
a
p
a
q
=
a
q
a
r
a
p
a
q
−1=
a
q
a
r
−1
a
r
−a
q
a
q
−a
p
=
a
q
a
p
a+(r−1)d−[a+(q−1)d]
a+(q−1)d−[a+(p−1)d]
=
a+(q−1)d
a+(p−1)d
p−q
q−r
=
a
q
a
p
a
q
a
r
=
a
r
a
s
a
q
a
r
−1=
a
r
a
s
−1
a
q
−a
r
a
r
−a
s
=
a
q
a
r
a+(q−1)d−[a+(r−1)d]
a+(r−1)d−[a+(s−1)d]
=
a+(q−1)d
a+(r−1)d
q−r
r−s
=
a
p
a
q
From (1) and (2)
a
q
a
p
=
a
p
a
q
p−q
q−r
=
q−r
r−s
Hence p−q,q−r,r−s are in GP
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