题目内容
先观察下列等式，然后用你发现的规律解答下列问题．
=1-
=
-
=
-
┅┅
（1）计算
+
+
+
+
=______；
（2）探究
+
+
+…+
=______；（用含有n的式子表示）
（3）若
+
+
+…+
的值为
，求n的值．
| 1 |
| 1×2 |
| 1 |
| 2 |
| 1 |
| 2×3 |
| 1 |
| 2 |
| 1 |
| 3 |
| 1 |
| 3×4 |
| 1 |
| 3 |
| 1 |
| 4 |
┅┅
（1）计算
| 1 |
| 1×2 |
| 1 |
| 2×3 |
| 1 |
| 3×4 |
| 1 |
| 4×5 |
| 1 |
| 5×6 |
（2）探究
| 1 |
| 1×2 |
| 1 |
| 2×3 |
| 1 |
| 3×4 |
| 1 |
| n(n+1) |
（3）若
| 1 |
| 1×3 |
| 1 |
| 3×5 |
| 1 |
| 5×7 |
| 1 |
| (2n-1)(2n+1) |
| 17 |
| 35 |
（1）原式=1-
+
-
+
-
+
-
+
-
=1-
=
；
（2）原式=1-
+
-
+
-
+
-
+…+
-
=1-
=
；
（3）
+
+
+…+
=
(1-
)+
(
-
)+
(
-
)+…+
(
-
)
=
(1-
)=
由
=
，解得n=17，
经检验n=17是方程的根，
∴n=17．
| 1 |
| 2 |
| 1 |
| 2 |
| 1 |
| 3 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 4 |
| 1 |
| 5 |
| 1 |
| 5 |
| 1 |
| 6 |
| 1 |
| 6 |
| 5 |
| 6 |
（2）原式=1-
| 1 |
| 2 |
| 1 |
| 2 |
| 1 |
| 3 |
| 1 |
| 3 |
| 1 |
| 4 |
| 1 |
| 4 |
| 1 |
| 5 |
| 1 |
| n |
| 1 |
| n+1 |
| 1 |
| n+1 |
| n |
| n+1 |
（3）
| 1 |
| 1×3 |
| 1 |
| 3×5 |
| 1 |
| 5×7 |
| 1 |
| (2n-1)(2n+1) |
=
| 1 |
| 2 |
| 1 |
| 3 |
| 1 |
| 2 |
| 1 |
| 3 |
| 1 |
| 5 |
| 1 |
| 2 |
| 1 |
| 5 |
| 1 |
| 7 |
| 1 |
| 2 |
| 1 |
| 2n-1 |
| 1 |
| 2n+1 |
=
| 1 |
| 2 |
| 1 |
| 2n+1 |
| n |
| 2n+1 |
由
| n |
| 2n+1 |
| 17 |
| 35 |
经检验n=17是方程的根，
∴n=17．
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