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Question

# If the pth,qth & rth terms of an AP are in GP.Then the common ratio of the GP is

A
qrpq
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B
q+rqp
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C
rppq
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D
qrqp
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Solution

## The correct option is A q−rp−qLet first term of A.P is a and common difference is d∴Tp=a+(p−1)dTq=a+(q−1)dTr=a+(r−1)dare in G.PLet the first of G.P be x and common ratio be yThen a+(p−1)d=x ...(1)a+(q−1)d=xy ...(2)a+(r−1)d=xy2Subtract (1) from (2), we get (q−1)d−(p−1)d=xy−x⇒(q−p)d=xy−x ...(4)Subtract (2) from (3) (r−1)d−(q−1)d=xy2−xy=y(xy−x)⇒(r−q)d=y(xy−x) ...(5)Divide (4) by (5), we get(q−r)(p−q)=y

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