If q is the mean proportional between p and r, show that: pqr(p + q + r)3 = (pq + qr + rp)3. - Mathematics

If q is the mean proportional between p and r, show that: pqr(p + q + r)³ = (pq + qr + rp)³.

Sum

Given, q is the mean proportional between p and r.

⇒ q² = pr

L.H.S = pqr(p + q + r)³

= qq²(p + q + r)³

= q³(p + q + r)³ ...[∵ q² = pr]

= [q(p + q + r)]³

= (pq + q² + qr)³

= (pq + pr + qr)³ ...[∵ q² = pr]

= R.H.S

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