题目内容
观察下列成立的式子：
=1-
，
=
-
，
=
-
，
=
-
…
（1）则第n个算式为______=______．
（2）如果将上列式子左右相加得：
+
+
+
=1-
+
-
+
-
+
=1-
=
根据这个结果，则请你直接写出下列式子的结果：①
+
+
+…+
=______；
②
+
+
+…+
=______；
（3）探究并计算
+
+
+
+
+
+
+
+
．
| 1 |
| 1×2 |
| 1 |
| 2 |
| 1 |
| 2×3 |
| 1 |
| 2 |
| 1 |
| 3 |
| 1 |
| 3×4 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 4×5 |
| 1 |
| 4 |
| 1 |
| 5 |
（1）则第n个算式为______=______．
（2）如果将上列式子左右相加得：
| 1 |
| 1×2 |
| 1 |
| 2×3 |
| 1 |
| 3×4 |
| 1 |
| 4×5 |
| 1 |
| 2 |
| 1 |
| 2 |
| 1 |
| 3 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 5 |
| 1 |
| 5 |
| 4 |
| 5 |
| 1 |
| 1×2 |
| 1 |
| 2×3 |
| 1 |
| 3×4 |
| 1 |
| 2008×2009 |
②
| 1 |
| 1×2 |
| 1 |
| 2×3 |
| 1 |
| 3×4 |
| 1 |
| n×(n+1) |
（3）探究并计算
| 1 |
| 2 |
| 1 |
| 6 |
| 1 |
| 12 |
| 1 |
| 20 |
| 1 |
| 30 |
| 1 |
| 42 |
| 1 |
| 56 |
| 1 |
| 72 |
| 1 |
| 90 |
（1）观察给出的式子可知：第n个算式为
=
-
（n为正整数）；
（2）①
+
+
+…+
=1-
+
-
+
-
+…+
-
=1-
=
；
②
+
+
+…+
=1-
+
-
+
-
+…+
-
=1-
=
；
（3）
+
+
+
+
+
+
+
+
=
+
+
+
+
+
+
+
+
=1-
+
-
+
-
+…+
-
=1-
=
．
故答案为：
；
．
| 1 |
| n(n+1) |
| 1 |
| n |
| 1 |
| n+1 |
（2）①
| 1 |
| 1×2 |
| 1 |
| 2×3 |
| 1 |
| 3×4 |
| 1 |
| 2008×2009 |
=1-
| 1 |
| 2 |
| 1 |
| 2 |
| 1 |
| 3 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 2008 |
| 1 |
| 2009 |
=1-
| 1 |
| 2009 |
=
| 2008 |
| 2009 |
②
| 1 |
| 1×2 |
| 1 |
| 2×3 |
| 1 |
| 3×4 |
| 1 |
| n×(n+1) |
=1-
| 1 |
| 2 |
| 1 |
| 2 |
| 1 |
| 3 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| n |
| 1 |
| n+1 |
=1-
| 1 |
| n+1 |
=
| n |
| n+1 |
（3）
| 1 |
| 2 |
| 1 |
| 6 |
| 1 |
| 12 |
| 1 |
| 20 |
| 1 |
| 30 |
| 1 |
| 42 |
| 1 |
| 56 |
| 1 |
| 72 |
| 1 |
| 90 |
=
| 1 |
| 1×2 |
| 1 |
| 2×3 |
| 1 |
| 3×4 |
| 1 |
| 4×5 |
| 1 |
| 5×6 |
| 1 |
| 6×7 |
| 1 |
| 7×8 |
| 1 |
| 8×9 |
| 1 |
| 9×10 |
=1-
| 1 |
| 2 |
| 1 |
| 2 |
| 1 |
| 3 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 9 |
| 1 |
| 10 |
=1-
| 1 |
| 10 |
=
| 9 |
| 10 |
故答案为：
| 2008 |
| 2009 |
| n |
| n+1 |
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，
，
，
…
根据这个结果，则请你直接写出下列式子的结果：①
=______；
+
+
+…+
=______；
．