现在用中文快速解答这三个题目，要求解答过程简单，视频长度控制在3分钟之内---Here is the extraction of the content from the image: **Problem 16.** (15 分) 设数列$\{a_n\}$满足 $a_1=3$, $\frac{a_{n+1}}{n+1} = \frac{a_n}{n} + \frac{1}{n(n+1)}$. (1) 证明: $\{ \frac{a_n}{n} \}$ 为等差数列; (2) 设$f(x) = a_1x + a_2x^2 + \dots + a_mx^m$, 求$f'(-2)$. **Problem 17.** (15 分) 如图所示的四棱锥 $P-ABCD$ 中, $PA \perp$ 平面 $ABCD$, $BC \parallel AD$, $AB \perp AD$. (1) 证明: 平面 $PAB \perp$ 平面 $PAD$; (2) 若 $PA=AB=\sqrt{2}$, $AD=\sqrt{3}+1$, $BC=2$, $P, B, C, D$ 在同一个球面上, 设该球面的球心为 $O$. (i) 证明: $O$ 在平面 $ABCD$ 上; (ii) 求直线 $AC$ 与直线 $PO$ 所成角的余弦值. **Geometric Figure Description for Problem 17:** * Type: 3D Geometric Figure (Pyramid) * Main Elements: * Vertices: P, A, B, C, D. * Base: Quadrilateral ABCD. * Apex: P. * Edges: PA, PB, PC, PD, AB, BC, CD, AD. * Lines: PA is shown as a dashed line indicating it is perpendicular to the base. AB, BC, CD, AD are the edges of the base. PB, PC, PD are the slant edges. * Right Angle Symbol: A small square is shown at vertex A between lines AB and AD, indicating that angle BAD is a right angle ($AB \perp AD$). * Parallel Lines: BC and AD are shown as parallel. * Orientation: The base ABCD is shown in perspective, appearing roughly diamond-shaped. The vertex A is in the foreground-left, B is to the right of A, D is to the right and slightly behind A, and C is between B and D and further back. P is vertically above A. * Labeling: All vertices A, B, C, D, P are labeled. **Problem 18.** (17 分) 已知椭圆 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a>b>0)$ 的离心率为 $\frac{2\sqrt{2}}{3}$, 下顶点为 $A$, 右顶点为 $B$, $|AB| = \sqrt{10}$. (1) 求 $C$ 的方程; (2) 已知动点 $P$ 不在 $y$ 轴上, 点 $R$ 在射线 $AP$ 上, 且满足 $|AP| \cdot |AR| = 3$. (i) 设 $P(m,n)$, 求 $R$ 的坐标 (用 $m,n$ 表示); (ii) 设 $O$ 为坐标原点, $Q$ 是 $C$ 上的动点, 直线 $OR$ 的斜率是直线 $OP$ 的斜率的 3 倍, 求 $|PQ|$ 的最大值.