问题 解答题
观察下列计算:
1
2
+1
=
2
-1
(
2
+1)(
2
-1)
=
2
-1
1
3
+
2
=
3
-
2
(
3
+
2
)(
3
-
2
)
=
3
-
2
1
4
+
3
=
4
-
3
(
4
+
3
)(
4
-
3
)
=
4
-
3
;…
则:
(1)
1
10
+
9
=______,
1
100
+
99
=______;
(2)从计算结果找出规律:______;
(3)利用这一规律计算:
1
2
+1
+
1
3
+
2
+
1
4
+
3
+…+
1
2006
+
2005
)(
2006
+1
)的值.
答案

(1)

1
10
+
9

=

10
-
9
(
10
+
9
)(
10
-
9
)

=

10
-
9
10-9

=

10
-
9

1
100
+
99

=

100
-
99
(
100
+
99
)(
100
-
99
)

=

100
-
99
100-99

=

100
-
99

(2)

1
n+1
+
n
=
n+1
-
n
(n是正整数)

(3)(

1
2
+1
+
1
3
+
2
+
1
4
+
3
+…+
1
2006
+
2005
)(
2006
+1

=【(

2
-1)+(
3
-
2
)+(
4
-
3
)
+…+(
2006
-
2005
)】(
2006
+1

=(

2
-1+
3
-
2
+
4
-
3
+
2006
-
2005
)(
2006
+1

=(

2006
-1)(
2006
+1

=2006-1

=2005

判断题
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