定义 $x*y=\dfrac{x+y}{1+xy}$，则 $(\cdots((2*3)*4)\cdots)*21=$ （ ）
A．$\dfrac 56$
B．$\dfrac 65$
C．$\dfrac{115}{116}$
D．$\dfrac{116}{115}$
答案 D．
解析 令 $x=\dfrac {a+1}{a-1}$，$y=\dfrac{b+1}{b-1}$，则有\[x*y=\dfrac{\dfrac{a+1}{a-1}+\dfrac{b+1}{b-1}}{1+\dfrac{a+1}{a-1}\cdot \dfrac{b+1}{b-1}}=\dfrac{ab+1}{ab-1},\]而 $x=\dfrac{a+1}{a-1}$ 即 $a=\dfrac{x+1}{x-1}$，从而\[(\cdots((2*3)*4)\cdots)*21=\dfrac{\displaystyle\prod_{k=2}^{21}\dfrac{k+1}{k-1}+1}{\displaystyle\prod_{k=2}^{21}\dfrac{k+1}{k-1}-1}=\dfrac{\dfrac{21\cdot 22}{1\cdot 2}+1}{\dfrac{21\cdot 22}{1\cdot 2}-1}=\dfrac{116}{115}.\]