Question
If pth,qth,rth and sth terms of an A.P. be in G.P., then (p - q),(q - r),(r - s) will be in
Answer: Option A
:
A
If a and d be the first term and common difference of theA.P.
Then Tp = a + (p - 1)d, Tq = a + (q - 1)d and
Tr = a+(r - 1)d.
If Tp,Tq,Tr are in G.P.
Then its common ratio R = TqTp = TrTq = Tq−TrTp−Tq
= [a+(q−1)d]−[a+(r−1)d][a+(p−1)d]−[a+(q−1)d] = q−rp−q
Similarly, we can show that R =q−rp−q = r−sq−r
Hence (p - q), (q - r),(r - s) be in G.P.
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:
A
If a and d be the first term and common difference of theA.P.
Then Tp = a + (p - 1)d, Tq = a + (q - 1)d and
Tr = a+(r - 1)d.
If Tp,Tq,Tr are in G.P.
Then its common ratio R = TqTp = TrTq = Tq−TrTp−Tq
= [a+(q−1)d]−[a+(r−1)d][a+(p−1)d]−[a+(q−1)d] = q−rp−q
Similarly, we can show that R =q−rp−q = r−sq−r
Hence (p - q), (q - r),(r - s) be in G.P.
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