问题 解答题
阅读下列解题过程:
1
2
+1
=
1×(
2
-1)
(
2
+1)×(
2
-1)
=
(
2
-1)
(
2
)
2
-12
=
2
-1
1
3
+
2
=
1×(
3
-
2
)
(
3
+
2
)×(
3
-
2
)
=
3
-
2
(
3
)
2
-(
2
)
2
=
3
-
2
1
4
+
3
=
1×(
4
-
3
)
(
4
+
3
)×(
4
-
3
)
=
4
-
3
(
4
)
2
-(
3
)
2
=
4
-
3

请回答下列问题:
(1)观察上面的解题过程,请直接写出式子
1
n
+
n-1
=______;(n为整数,且n>1)
(2)利用上面所提供的解法,请化简
1
2
+1
+
1
3
+
2
+
1
4
+
3
+…+
1
2012
+
2011
+
1
2013
+
2012
的值.
答案

(1)

1
n
+
n-1
=
n
-
n-1
(
n+
n-1
)(
n
-
n-1
)
=
n
-
n-1

故答案为:

n
-
n-1

(2)原式=

2
-1+
3
-
2
+
4
-
3
+…+
2012
-
2011
+
2013
-
2012
=
2013
-1

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