If q is the mean proportional between p and r, show that :

pqr(p + q + r)³ = (pq + qr + pr)³.

Since, q is the mean proportional between p and r

$\therefore \dfrac{p}{q} = \dfrac{q}{r} \Rightarrow q^2 = pr.$

Substituting pr = q² in L.H.S. of pqr(p + q + r)³ = (pq + qr + pr)³

⇒ q.q²(p + q + r)³

⇒ q³(p + q + r)³

⇒ [q(p + q + r)]³

⇒ (pq + q² + qr)³

⇒ (pq + pr + qr)³ = R.H.S. (As q² = pr).

Hence, proved that pqr(p + q + r)³ = (pq + qr + pr)³.