.prove that `(a^(p+q))^(2)(a^(q+r))^(2)(a^(r+p))^(2)/((a^(p)*a^(q)*a^(r))^(4))=1`

if q is the mean proportional between p and r , prove that : p^(2) - q^(2) + r^(2) = q^(4) ((1)/(p^(2))-(1)/(q^(2)) + (1)/(r^(2))) .

Simply (a^(p)/a^(q))^(p+q-r).(a^(q)/a^(r ))^(q+r-p).(a^(r )/a^(p))^(r+p-q)

If p,q,r are in A.P the p^(2)(q+r),q^(2)(r+p),r^(2)(p+q) are in

If p, q, r are in A.P the p^(2) (q + r), q^(2) (r + p), r^(2) (p+q) are in

Prove that |A| = | ((q+r)^2 ,p^2,p^2),(q^2 ,(r + p)^2, q^2),(r^2 , r^2 , (p+q)^2)| = 2pqr(p+q+r)^3

Prove that (p+q omega+r omega^(2))/(r+p omega+q omega^(2))=omega^(2)

If (log x)/(q-r)=(log y)/(r-p)=(log z)/(p-q) prove that x^(q+r)*y^(r+p)*z^(p+q)=x^(p)*y^(q).z^(r)

If p, q, r, s are in G.P. then show that (q -r )^(2) + ( r -p) ^(2) + ( s-q) ^(2) = ( p -s) ^(2)

if p: q :: r , prove that p : r = p^(2): q^(2) .