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Question
If p, q and r in continued proportion, then prove the following:
(p + q + r )(p - q + r) = p2 + q2 + r2
Solution
p : q :: q : r ⇒ q2 = pr
(p + q + r )(p - q + r) = p2 + q2 + r2
LHS
(p + q + r)(p - q + r)
= p2 + pq + pr - pq - q2 - qr + pr + qr + r2
= p2 + 2pr - q2 + r2
= p2 + 2q2 - q2 + r2
= p2 + q2 + r2 = RHS
LHS = RHS. Hence, proved.
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