Prove that `=|a cc-a a+cc bb-c b+c a-bb-c0a-c x y z1+x+y|=0` implies that `a ,b ,c` are in A.P. or `a ,c ,b` are in G.P.

Prove that: |1a a^2-b c1bb^2-c a1cc^2-a b|=0

If a^x=b^y=c^z and a,b,c are in G.P. show that 1/x,1/y,1/z are in A.P.

If a,b,c are in A.P and a,b,d are in G.P, prove that a,a-b,d-c are in G.P.

The determinant |a b aalpha+bb c balpha+c aalpha+bbalpha+c0|=0, if a ,b , c are in A.P. a ,b ,c are in G.P. a ,b ,c are in H.P. alpha is a root of the equation a x^2+b c+c=0 (x-alpha) is a factor of a x^2+2b x+c

If x+y+z=0 prove that |a x b y c z c y a z b x b z c x a y|=x y z|a b cc a bb c a|

If a, b, c are in A.P., b, c, d are in G.P. and 1/c ,""1/d ,""1/e are in A.P. prove that a, c, e are in G.P.