Монгол Бодлогын Сан

Заавар: $$\dfrac{1}{n(n+2)}=\dfrac{1}{2}\left(\dfrac{1}{n}-\dfrac{1}{n+2}\right)$$

Бодолт: \begin{align*} \sum&=1+\displaystyle\frac{1}{2\cdot 4}+\displaystyle\frac{1}{4\cdot 6}+\dots +\displaystyle\frac{1}{20\cdot 22}\\ &=1+\dfrac12\Big(\dfrac12-\dfrac14\Big)+\dfrac12\Big(\dfrac14-\dfrac16\Big)+\dots+\dfrac12\Big(\dfrac{1}{20}-\dfrac{1}{22}\Big)\\ &=1+\dfrac12\cdot\dfrac12+\dfrac12\Big(-\dfrac14+\dfrac14\Big)+\dots+\dfrac12\Big(-\dfrac{1}{20}+\dfrac{1}{22}\Big)-\dfrac12\cdot\dfrac{1}{22}\\ &=1+\dfrac12\Big(\dfrac12-\dfrac{1}{22}\Big)=1+\dfrac12\cdot\dfrac{5}{11}=\dfrac{27}{22} \end{align*}