每日一题[2285]分拆

$\dfrac{4}{1 \cdot 2 \cdot 3}+\dfrac{5}{2 \cdot 3 \cdot 4}+\cdots+\dfrac{n+3}{n(n+1)(n+2)}=$ [[nn]].

答案    $\dfrac{n(11+5n)}{4(n+1)(n+2)}$．

解析    设题中代数式为 $m$, 则

$$\begin{split} m&=\sum_{k=1}^n\left(\dfrac{\frac 32}{k}-\dfrac2{k+1}+\dfrac{\frac 12}{k+2}\right)\\ &=\sum_{k=1}^n\dfrac{\frac 32}{k}-\sum_{k=2}^{n+1}\dfrac{2}{k}+\sum_{k=3}^{n+2}\dfrac{\frac 12}{k}\\ &=2-\dfrac2{n+1}-\dfrac 12-\dfrac 14+\dfrac{1}{2(n+1)}+\dfrac{1}{2(n+2)}\\ &=\dfrac{n(11+5n)}{4(n+1)(n+2)}.\end{split}$$