问题 解答题
阅读下面问题:
1
1+
2
=
1×(
2
-1)
(
2
+1)(
2
-1)
=
2
-1

1
3
+
2
=
3
-
2
(
3
+
2
)(
3
-
2
)
=
3
-
2

1
5
+2
=
5
-2
(
5
+2)(
5
-2)
=
5
-2

试求:(1)
1
n+1
+
n
(n为正整数)的值.
(2)利用上面所揭示的规律计算:
1
1+
2
+
1
2
+
3
+
1
3
+
4
+…+
1
2008
+
2009
+
1
2009
+
2010
答案

(1)

1
n+1
+
n
=
n+1
-
n

(2)

1
1+
2
+
1
2
+
3
+
1
3
+
4
+…+
1
2008
+
2009
+
1
2009
+
2010

=

2
-1+
3
-
2
+
4
-
3
+…+
2009
-
2008
+
2010
-
2009

=

2010
-1.

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