$解:由题意可得，\frac{M}{a+1}=\frac{a^{2} }{a(a+1)}=\frac{a}{a+1}，$
$则M=a，\ $
$那么\frac{a}{a+1}- \frac{1}{a²+a} = \frac{a^{2} }{a(a+1)} -\frac{1}{a(a+1)}\ $
$=\frac{a²-1}{a(a+1)}= \frac{(a+1)(a-1)}{a(a+1)}=\frac{a-1}{a}.\ $
$当a=100时，原式=\frac{100-1}{100}=\frac{99}{100}.$
