Mon December 25th 2023

In triangle $OAB$, the $\overrightarrow{OA}$ = 3i + 4k and the vector $\overrightarrow{OB}$ = i + 2j - 2k. Find $\overrightarrow{OP}$, where $P$ is the point where the bisector of $\angle AOB$ intersects $AB$.

$\frac{1}{1\cdot 2\cdot 3}+\frac{1}{2\cdot 3\cdot 4}+\frac{1}{n(n+1)(n+2)}=\frac{n\left(n+3\right)}{4\left(n+1\right)\left(n+2\right)}$

$(\frac{1}{2}-\frac{1}{2}+\frac{1}{4})+(\frac{1}{6}-\frac{1}{3}+\frac{1}{6})+(\frac{1}{8}-\frac{1}{4}+\frac{1}{8})+(\frac{1}{10}-\frac{1}{5}+\frac{1}{10})+\cdots +\frac{1}{2(k+1)}-\frac{1}{k+1}+\frac{1}{2(k+2)}$