RU-ALLRUSSIAN 1978 Problem 17

Let $\mathbf{x}_{\mathrm{n}} = (1 + \sqrt{2} + \sqrt{3})^{\mathrm{n}}$ . We may write $\mathbf{x}_{\mathrm{n}} = \mathbf{a}_{\mathrm{n}} + \mathbf{b}_{\mathrm{n}}\sqrt{2} + \mathbf{c}_{\mathrm{n}}\sqrt{3} + \mathbf{d}_{\mathrm{n}}\sqrt{6}$ , where $\mathbf{a}_{\mathrm{n}}, \mathbf{b}_{\mathrm{n}}, \mathbf{c}_{\mathrm{n}}, \mathbf{d}_{\mathrm{n}}$ are integers. Find the limit as $\mathbf{n}$ tends to infinity of $\mathbf{b}_{\mathrm{n}} / \mathbf{a}_{\mathrm{n}}, \mathbf{c}_{\mathrm{n}} / \mathbf{a}_{\mathrm{n}}, \mathbf{d}_{\mathrm{n}} / \mathbf{a}_{\mathrm{n}}$ .