Question

if a, b, c are in g.p prove that a²+b², a b+b c, b²+c² are also in g.p.

Answer 1

if a b c are in gp it means b=k*a, c=k*b ,and c=k^2a, where k is some number. so

a^2+b^2 =a^2(1+k^2),

ab+bc=ka^2+k^3a^2=a^2*k*(k^2+1)

b^2+c^2=k^2a^2+k^4a^2=a^2*k^2*(k^2+1)

clearly in gp