Prove that `a^p b^ q <((a p+b q)/(p+q))^(p+q)dot`

If the p^("th"), q^("th") and r^("th") terms of a G.P. are a, b and c, respectively. Prove that a^(q – r) b^(r-p) c^(P - q) = 1.

Prove that p to (~q vee r) -= ~p vee (~q vee r) using truth table.

Prove that p + q = 0 " if " x^(3) - 3px + 2q " is divisible by" (x + 1)^(2)

Prove that p x^(q-r)+q x^(r-p)+r x^(p-q)> p+q+r ,w h e r ep ,q ,r are distinct and x!=1.

If the angles A,B,C of a triangle are in A.P. and sides a,b,c, are in G.P., then prove that a^2, b^2,c^2 are in A.P.

Sum of the first p, q and r terms of an A.P. are a, b and c, respectively. Prove that a/p(q-r)+b/q (r-p) +c/r (p-q)=0

Prove that q to p -= ~p to ~q

Prove that Q to P -= P to - Q

Prove that ~((~p)^^q) -=pvv(~q) .

Prove that ~(~pto ~q) -=~p ^^q