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Question
Answer the following:
If p, q, r, s are in G.P., show that (p2 + q2 + r2) (q2 + r2 + s2) = (pq + qr + rs)2
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Solution
Let R be the common ratio of the G.P.
Then q = pR, r = pR2, s = pR3
∴ (p2 + q2 + r2)(q2 + r2 + S2)
= (p2 + p2R2 + p2R4)(p2R2 + p2R4 + p2R6)
= p2(1 + R2 + R4)·p2R2(1 + R2 + R4)
= p4R2(1 + R2 + R4)2 ...(1)
and (pq +qr + rs)2 = [p(pR) + (pR)(pR2) + (pR2)(pR3)]2
= (p2R + p2R3 + p2R5)2
= [p2R (1 + R2 + R4)]2
= p4R2 (1 + R2 + R4)2 ...(2)
From (1) and (2), we get,
(p2 + q2 + r2)(q2 + r2 + s2) = (pq + qr + rs)2
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