Question

If y is the mean proportional between X and z. Prove that xyz(X+y+z)^3=(xy+yz+zx)^3

Answer 1

mean proportional means :   x/y = y/z     =>   y² = zx

LHS = y y² (x+y +z)³ 
RHS = (xy + yz + y²)³ = y³ (x+z+y)³

so proved.

Answer 2

y is the mean proportional between x and z

so, y/z = x/y

y^2 =xz

now,

LHS =xyz(x + y + z)^3

=y(xz){x + y + z }^3

=y^3(x + y + z)^3

=(xy + yz + zx)^3 = RHS