问题 填空题
观察式子
1
1×3
=
1
2
(1-
1
3
)
1
3×5
=
1
3
(
1
3
-
1
5
)
1
5×7
=
1
2
(
1
5
-
1
7
)
,…由此可知
1
1×3
+
1
3×5
+
1
5×7
+…+
1
(2n-1)×(2n+1)
=______.
答案

原式=

1
2
(1-
1
3
)+
1
2
1
3
-
1
5
)+…+
1
2
1
2n-1
-
1
2n+1

=

1
2
(1-
1
3
+
1
3
-
1
5
+…+
1
2n-1
-
1
2n+1

=

1
2
(1-
1
2n+1

=

1
2
×
2n
2n+1

=

n
2n+1

故答案为

n
2n+1

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