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Question

# If the pth,qth,rth and sth terms of an A.P. are in G.P. then pâˆ’q,qâˆ’r,râˆ’s are in .........

A
A.P
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B
H.P
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C
G.P
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D
A.G.P
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Solution

## The correct option is B G.Ppth,qth,rth and sth terms of an A.P. are in G.P.Let a be first term and d be common difference of A.P. and R be common ratio of G.P.⇒a+(p−1)d,a+(q−1)d,a+(r−1)d,a+(s−1)d are in G.P.⇒R=a+(q−1)da+(p−1)d=a+(r−1)da+(q−1)d=a+(s−1)da+(r−1)dConsider a+(q−1)da+(p−1)d=a+(r−1)da+(q−1)dSubtracting 1 on both sides gives⇒a+(q−1)da+(p−1)d−1=a+(r−1)da+(q−1)d−1⇒(q−p)da+(p−1)d=(r−q)da+(q−1)d⇒R=q−rp−q -----(1)Similarly,R=r−sq−r -----(2)From (1) and (2)q−rp−q=r−sq−r∴p−q,q−r,r−s are in G.P.Hence, option C.

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