[ Systems of Linear Equations ]

Find all solutions of the system of equations $x\left(1-2^{-n}\right)+y\left(1-2^{-n-1}\right)+z\left(1-2^{-n-2}\right)=0, \quad$ where $n=1,2,3,4, \ldots$
The given infinite system of equations is equivalent to the system of two equations: $x+y+z=4 x+2 y+z=0$. Indeed, if we subtract from the first equation the second one multiplied by $2^{-n-2}$, we obtain the $n$-th equation of the original system.

On the other hand, from the first two equations of the original system $x(1-1 / 2)+y(1-1 / 4)+z(1-1 / 8)=0, x(1-1 / 4)+$ $y(1-1 / 8)+z\left(1-\frac{1}{16}\right)=0$ the specified two equations follow. Indeed, subtracting the second equation from the first, we get $x / 4+y / 8+z / 16=0$. Adding this to the second equation, we get $x+y+z=0$.

## Answer

$(t,-3 t, 2 t)$ ( $t$ - any number).

## Problem