题目内容
先阅读下列内容,然后解答问题因为
1 |
1×2 |
1 |
2 |
1 |
2×3 |
1 |
2 |
1 |
3 |
1 |
3×4 |
1 |
3 |
1 |
4 |
1 |
9×10 |
1 |
9 |
1 |
10 |
所以:
1 |
1×2 |
1 |
2×3 |
1 |
9×10 |
1 |
2 |
1 |
2 |
1 |
3 |
1 |
9 |
1 |
10 |
9 |
10 |
请计算:
①
1 |
1×2 |
1 |
2×3 |
1 |
2006×2007 |
②
1 |
1×3 |
1 |
3×5 |
1 |
5×7 |
1 |
2005×2007 |
分析:(1)分子为1,分母是两个连续自然数的乘积,第n项为
=
-
,所以原式=1-
+
-
+
-
+…+
-
=1-
=
;
(2)分子为1,分母是两个连续奇数的乘积,第n项为
=
(
-
),所以原式=
(1-
+
-
+
-…+
-
)=
(1-
)=
.
1 |
n×(n+1) |
1 |
n |
1 |
n+1 |
1 |
2 |
1 |
2 |
1 |
3 |
1 |
3 |
1 |
4 |
1 |
2006 |
1 |
2007 |
1 |
2007 |
2006 |
2007 |
(2)分子为1,分母是两个连续奇数的乘积,第n项为
1 |
n×(2n-1) |
1 |
2 |
1 |
n |
1 |
2n-1 |
1 |
2 |
1 |
3 |
1 |
3 |
1 |
5 |
1 |
5 |
1 |
2005 |
1 |
2007 |
1 |
2 |
1 |
2007 |
1003 |
2007 |
解答:解:①
+
+…+
=1-
+
-
+
-
+…+
-
=1-
=
;
②
+
+
+…+
=
(1-
+
-
+
-…+
-
)
=
(1-
)
=
.
1 |
1×2 |
1 |
2×3 |
1 |
2006×2007 |
=1-
1 |
2 |
1 |
2 |
1 |
3 |
1 |
3 |
1 |
4 |
1 |
2006 |
1 |
2007 |
=1-
1 |
2007 |
=
2006 |
2007 |
②
1 |
1×3 |
1 |
3×5 |
1 |
5×7 |
1 |
2005×2007 |
=
1 |
2 |
1 |
3 |
1 |
3 |
1 |
5 |
1 |
5 |
1 |
2005 |
1 |
2007 |
=
1 |
2 |
1 |
2007 |
=
1003 |
2007 |
点评:解决这类题目找出变化规律,消去中间项,只剩首末两项,使运算变得简单.
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