The x,y,z coordinates of each vertex of a triangle are in geometric progression  .The x and y coordinates of the centroid of the triangle are 1 and 2 respectively. The distance of the centroid from the origin is.

Given x,y,z coordinates of each vertex of a triangle are in geometric progressionand the centroid of the triangle is x1+x2+x33,y1+y2+y33,z1+z2+z33=(1,2,k)it implies that x1+x2+x3=3,y1+y2+y3=6suppose that r is common ratio of geometric progression, r(x1+x2+x3)=6⇒r(3)=6⇒r=2Hence, z1+z2+z3=r2(x1+x2+x3)=4(3)=12The centroid of the triangle is 1,2,4The distance of the centroid from origin is 1+4+16=21