$$\begin{aligned} & \mathrm{Sn}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20} \ldots . \mathrm{N} \text { terms } \\ & \mathrm{S}_{2025}=\sum_{\mathrm{n}=1}^{2025} \frac{1}{\mathrm{n}(\mathrm{n}+1)}=\sum_{\mathrm{n}=1}^{2025}\left(\frac{1}{\mathrm{n}}-\frac{1}{\mathrm{n}+1}\right) \\ & =\left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right) \quad \cdots \cdot\left(\frac{1}{2025}-\frac{1}{2026}\right) \end{aligned}$$

$$\begin{aligned} & =\frac{2025}{2026} \\ & \sqrt{2026 . \mathrm{S}_{2025}}=\sqrt{2025}=45 \\ & \text { Given : } \frac{6}{2}[-2 \mathrm{p}+(6-1) \mathrm{p}]=45 \\ & 9 \mathrm{p}=45 \\ & \mathrm{p}=5 \\ & \left|\mathrm{~A}_{20}-\mathrm{A}_{15}\right|=|-5+19 \times 5|-[-5+14 \times 5] \\ & =|90-65| \\ & =25 \end{aligned}$$