题目内容
(2012•平和县质检)观察下面的变形规律:
=1-
;
=
-
;
=
-
;…
解答下面的问题:
(1)若n为正整数,请你猜想
=
-
-
;
(2)证明你猜想的结论;
(3)求和:
+
+
+…+
.
1 |
1×2 |
1 |
2 |
1 |
2×3 |
1 |
2 |
1 |
3 |
1 |
3×4 |
1 |
3 |
1 |
4 |
解答下面的问题:
(1)若n为正整数,请你猜想
1 |
n(n+1) |
1 |
n |
1 |
n+1 |
1 |
n |
1 |
n+1 |
(2)证明你猜想的结论;
(3)求和:
1 |
1×2 |
1 |
2×3 |
1 |
3×4 |
1 |
2011×2012 |
分析:(1)观察规律可得:
=
-
;
(2)根据分式加减法的运算法则求解即可证得结论的正确性;
(3)利用上面的结论,首先原式可化为:1-
+
-
+
-
+…+
-
,继而可求得答案.
1 |
n(n+1) |
1 |
n |
1 |
n+1 |
(2)根据分式加减法的运算法则求解即可证得结论的正确性;
(3)利用上面的结论,首先原式可化为:1-
1 |
2 |
1 |
2 |
1 |
3 |
1 |
3 |
1 |
4 |
1 |
2011 |
1 |
2012 |
解答:解:(1)
=
-
;
(2)
-
=
-
=
=
;
(3)
+
+
+…+
=1-
+
-
+
-
+…+
-
=1-
=
.
1 |
n(n+1) |
1 |
n |
1 |
n+1 |
(2)
1 |
n |
1 |
n+1 |
n+1 |
n(n+1) |
n |
n(n+1) |
n+1-n |
n(n+1) |
1 |
n(n+1) |
(3)
1 |
1×2 |
1 |
2×3 |
1 |
3×4 |
1 |
2011×2012 |
1 |
2 |
1 |
2 |
1 |
3 |
1 |
3 |
1 |
4 |
1 |
2011 |
1 |
2012 |
1 |
2012 |
2011 |
2012 |
点评:此题考查了分式的加减运算法则.此题难度适中,解题的关键是仔细观察,得到规律
=
-
,然后利用规律求解.
1 |
n(n+1) |
1 |
n |
1 |
n+1 |
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