∵当n=1时,
1
1×2 =1-
1
2 ,
当n=2时,
1
2×3 =
1
2 -
1
3 ,
当n=3时,
1
3×4 =
1
3 -
1
4 ,
…
∴当n=n时,
1
n(n-1) =
1
n -
1
n+1 .
∴
a
1×2 +
a
2×3 +
a
3×4 +…+
a
2010×2011
=a(
1
1×2 +
1
2×3 +
1
3×4 +…+
1
2010×2011 )
=a(1-
1
2 +
1
2 -
1
3 +
1
3 -
1
4 +…+
1
2010 -
1
2011 )
=a(1-
1
2011 )
=
2010
2011 a,
当a=2011时,原式=
2010
2011 ×2011=2010.