S=1^2\1*3+2^2\3*5+3^2\5*7+.+1000^2\1999*2001,则S=

解 ：归纳出这些项的 通式 即 为
n^2/(2n-1)(2n+1)=1/4+1/4(1/(2n-1)(2n+1) )=1/4+1/8(1/(2n-1)-1/(2n+1))
所以 S=1^21*3+2^23*5+3^25*7+.+1000^21999*2001
=1/4+1/8(1-1/3)+1/4+1/8(1/3-1/5)+1/4+1/8(1/5-1/7)+……+1/4+1/8(1/1999-1/2001)
=(1/4)*1000+1/8(1-1/2001)
=250+250/2001=250*2002/2001