RU-ALLRUSSIAN 1978 Problem 16

$\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots , \mathbf{a}_{\mathrm{n}}$ are real numbers. Let $\mathbf{b}_{\mathrm{k}} = (\mathbf{a}_{1} + \mathbf{a}_{2} + \ldots +\mathbf{a}_{\mathrm{k}}) / \mathbf{k}$ for $\mathbf{k} = 1, 2, \ldots , \mathbf{n}$ . Let $\mathbf{C} = (\mathbf{a}_{1} - \mathbf{b}_{1})^{2} + (\mathbf{a}_{2} - \mathbf{b}_{2})^{2} + \ldots +(\mathbf{a}_{\mathrm{n}} - \mathbf{b}_{\mathrm{n}})^{2}$ , and $\mathbf{D} = (\mathbf{a}_{1} - \mathbf{b}_{\mathrm{n}})^{2} + (\mathbf{a}_{2} - \mathbf{b}_{\mathrm{n}})^{2} + \ldots +(\mathbf{a}_{\mathrm{n}} - \mathbf{b}_{\mathrm{n}})^{2}$ . Show that $\mathbf{C} \leq \mathbf{D} \leq 2\mathbf{C}$ .