Question

If $x,y,z$ are in G.P. then $x^2+y^2,xy+yz,y^2+z^2$ are in?

Answer 1

x , y , z are in G.P.

$\Rightarrow y^2=xz$

Then $x^2+y^2,xy+yz,y^2+z^2$

$\Rightarrow x^2+xz,y(x+z),xz+z^2$

$\Rightarrow x(x+z),y(x+z),z(x+z)$

So $\left[y\right(x+z){]}^2=x(x+z\left)z\right(x+z)$

$\Rightarrow y^2(x+z{)}^2=xz(x+z{)}^2$

$\therefore x^2+y^2,xy+yz,y^2+z^2$

are also in GP.