题目内容
先观察下列各式：
=
(1-
)，
=
(
-
)，
=
(
-
)，…
=
(
-
)．
根据以上的观察，计算：
+
+
+…+
的值．
| 1 |
| 1×4 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 4×7 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 7 |
| 1 |
| 7×10 |
| 1 |
| 3 |
| 1 |
| 7 |
| 1 |
| 10 |
| 1 |
| n(n+3) |
| 1 |
| 3 |
| 1 |
| n |
| 1 |
| n+3 |
根据以上的观察，计算：
| 1 |
| 1×4 |
| 1 |
| 4×7 |
| 1 |
| 7×10 |
| 1 |
| 2005×2008 |
分析：由式
=
(1-
)，
=
(
-
)，
=
(
-
)，…
=
(
-
)可以看出，分子是1，分母是相差3的两个自然数的乘积等于以这两个自然数为分母，分子是1的两个分数差的
，由此规律把原算式裂项，求得结果即可．
| 1 |
| 1×4 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 4×7 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 7 |
| 1 |
| 7×10 |
| 1 |
| 3 |
| 1 |
| 7 |
| 1 |
| 10 |
| 1 |
| n(n+3) |
| 1 |
| 3 |
| 1 |
| n |
| 1 |
| n+3 |
| 1 |
| 3 |
解答：解：原式=
×（1-
）+
×（
-
）+
×（
-
）+…+
×（
-
）
=
×（1-
+
-
+
-
+…+
-
）
=
×（1-
）
=
×
=
．
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 7 |
| 1 |
| 3 |
| 1 |
| 7 |
| 1 |
| 10 |
| 1 |
| 3 |
| 1 |
| 2005 |
| 1 |
| 2008 |
=
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 4 |
| 1 |
| 7 |
| 1 |
| 7 |
| 1 |
| 10 |
| 1 |
| 2005 |
| 1 |
| 2008 |
=
| 1 |
| 3 |
| 1 |
| 2008 |
=
| 1 |
| 3 |
| 2007 |
| 2008 |
=
| 669 |
| 2008 |
点评：此题考查利用分数的拆项，把有理数的混合运算，转化为有规律的简便运算，使计算简便可行．
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