Question

The numbers $5x-y,2x+y$ and $x+2y$ form an arithmetic progression and the numbers $(y+1{)}^2,xy+1,(x-1{)}^2$ form a geometric progression. Find $x$ and $y$.

Answer 1

$5x-y,2x+y,x+2y$ forms an $A.P$

$⟹2(2x+y)=5x-y+x+2y⟹y=2x$

$(y+1{)}^2,xy+1,(x-1{)}^2$ forms a $G.P$

$⟹(xy+1{)}^2=(x-1{)}^2(y+1{)}^2⟹(2x^2+1{)}^2=(x-1{)}^2(2x+1{)}^2$

$⟹4x^3+7x^2-2x=0⟹x=0,-2,\frac{1}{4}⟹y=0,-4,\frac{1}{2}$