If `a, b, c` are in `G. P.`, prove that `a^2+ b^2, ab + bc, b^2 +c^2` are also in `G.P`

If a, b, c are in GP, prove that (a^(2)+b^(2)), (ab+bc), (b^(2)+c^(2)) are in GP.

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

If a, b, c are in G.P. then show that b^(2 ) = a.c.

If a,b,c are in A.P., prove that a^(2)+c^(2)-2bc=2a(b-c) .

If a,b,c,d are in G.P.,prove that a+b,b+c,c+d are also in G.P.

If a,b,c,d are in G.P.,prove that (a^(2)-b^(2)),(b^(2)-c^(2)),(c^(2)-d^(2)) are in G.P.and (1)/(a^(2)+b^(2)),(1)/(b^(2)+c^(2)),(1)/(c^(2)+d^(2)) are in G.P

If ab+bc+ca!=0 and a,b,c are in A.P. prove that a^(2)(b+c),b^(2)(c+a),c^(2)(a+b) are also in A.P.

If a,b,c are in G.P., then show that a(b-c)^2=c(a-b)^2

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

If a,b,c,d are in G.P.prove that: (a^(2)+b^(2)),(b^(2)+c^(2)),(c^(2)+d^(2)) are in G.P.(a^(2)-b^(2)),(b^(2)-c^(2)),(c^(2)-d^(2)) are in G.P.(1)/(a^(2)+b^(2)),(1)/(b^(2)+c^(2)),(1)/(c^(2)+d^(2)) are in G.P.(a^(2)+b^(2)+c^(2)),(ab+bc+cd),(b^(2)+c^(2)+d^(2))