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Question

# If p,q,r,s ϵ N and they are four consecutive terms of an AP, then the pth, qth, rth, sth terms of a GP are in

A
AP
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B
GP
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C
HP
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D
none of these
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Solution

## The correct option is B GPp,q,r,s are in AP⇒2q=p+r and 2r=s+q⇒2r=s+p+r2⇒s=3r−p2Now, the pth,qth,rth,sth terms of a GP can be written as atp−1,atq−1,atr−1,ats−1=atp−1,atp+r2−1,atr−1,ats−1=atp−1,atp−1tr−p2,atp−1tr−p,at3r−p2−1=atp−1,atp−1tr−p2,atp−1tr−p,atp−1t3(r−p)2=A,AR,AR2,AR3 where A=atp−1 and R=tr−p2Thus, the terms are in GP.∴ Ans. is option B.

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