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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
G.P
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C
H.P
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D
None of the above
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Solution

## The correct option is B G.PLet pth terms =a+(p−1)d,qth terms =a+(q−1)d,rth terms =a+(r−1)d,sth terms =a+(s−1)d,∴ Let x and y be the 1st term and common ratio of the G.P (As given) receptively.So, a+(p−1)d=x ...(1)a+(q−1)d=xy ...(2)a+(r−1)d=xy2 ...(3)a+(s−1)d=xy3 ...(4)Apply (1) - (2)a+pd−d−a−q−qd+d=x−xy(p−q)d=x(1−y) ...(5)Apply (2)-(3), we get(q−r)d=xy(1−y) ...(6)Apply (3) - (4)(r−s)d=xy2(1−y) ...(7)Now, (6) ÷ (5)(q−r)d(p−q)d=xy(1−y)x(1−y)(q−rp−q)=y ...(8)And, (7) ÷ (6}(r−s)d(q−r)d=xy2(1−y)xy(1−y)(r−3q−2)=y ...(9)from, (8) and (9)we get,(q−r)(p−q)=(r−s)(q−r)∴ Hence,(p−q),(q−r),(r−s) are in G.P.

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