Question

Prove que, se (p,q,r) é uma P.A, então (p²qr, pq²r, pqr²) também é.​

Answer 1

$\displaystyle \sf \text{Sabemos que numa P.A {\'e} v{\'a}lido} : \\\\ A_2 = \frac{A_1+A_3}{2} \\\\\ Temos : \\\\\ \text{Prove que, se (p,q,r) {\'e} uma P.A, ent{\~a}o} \ (p^2qr,pq^2r,pqr^2 ) \ \text{tamb{\'em} {\'e}}:$

Façamos :

$\displaystyle \sf (p,q,r) \to A_1 = p \ , \ A_2=q\ , \ A_3 = r \ \\\\\\ q = \frac{p+r}{2} \to \boxed{\sf p+r = 2q }\\\\\\ (p^2qr,pq^2r, pqr^2) \ \to A_1 = p^2qr\ , \ A_2 = pq^2r\ , \ A_3 = pqr^2 \\\\\\ pq^2r= \frac{p^2qr+pqr^2}{2} \\\\\\ pqr\cdot q =\frac{pqr(p+r) }{2} \\\\\\ \frac{p+r}{2} = \frac{pqr\cdot q}{pqr} \\\\\\ \huge\boxed{\sf p+r = 2 q\ }\checkmark \to (C.Q.D)$