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Question

If the pth,qth,rth and sth terms of an A.P. are in G.P,. then pq,qr,rs 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.P
Let pth terms =a+(p1)d,

qth terms =a+(q1)d,

rth terms =a+(r1)d,

sth terms =a+(s1)d,

Let x and y be the 1st term and common ratio of the G.P (As given) receptively.

So, a+(p1)d=x ...(1)

a+(q1)d=xy ...(2)

a+(r1)d=xy2 ...(3)

a+(s1)d=xy3 ...(4)

Apply (1) - (2)

a+pddaqqd+d=xxy

(pq)d=x(1y) ...(5)

Apply (2)-(3), we get

(qr)d=xy(1y) ...(6)

Apply (3) - (4)

(rs)d=xy2(1y) ...(7)

Now, (6) ÷ (5)

(qr)d(pq)d=xy(1y)x(1y)

(qrpq)=y ...(8)

And, (7) ÷ (6}

(rs)d(qr)d=xy2(1y)xy(1y)

(r3q2)=y ...(9)

from, (8) and (9)

we get,

(qr)(pq)=(rs)(qr)

Hence,

(pq),(qr),(rs) are in G.P.

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