If a,b,c are respectively the `p^(th),q^(th),r^(th)` terms of an A.P.,then prove that `a(q-r)+b(r-p)+c(p-q)=0`

If a,b,c are respectively the sums of p,q,r terms of an A.P., prove that a/p(q-r)+b/q(r-p)+c/r(p-q)=0

If x,y,z are positive and are the pth, qth and rth terms of a G.P. then prove that |[logx,p,1],[logy,q,1],[logz,r,1]|=0

If P(A/B) gt P(A) then prove that P(B/A) gt P(B)

If a^2,b^2,c^2 are in A.P. prove that 1/(b+c),1/(c+a),1/(a+b) are in A.P.

If the side lengths a,b and c are in A.P. then prove that cos ((A-C)/2) = 2sin (B/2)

If a^2,b^2,c^2 being A.P prove that cotA,cotB,cotC are also In A.P.

If x^(p)y^(q)=(x+y)^(p+q) , then prove that (dy)/(dx)=(y)/(x) .

If (a-c)^2,(c-a)^2,(a-b)^2 are in A.P.,prove that 1/(b-c),1/(c-a),1/(a-b) are in A.P.

The Inverse of the proposition (p ^^ ~~q)rarr r is: