问题 解答题
观察下列各式及验证过程:
1
2
(
1
3
-
1
4
)
=
1
3
3
8
验证:
1
2
(
1
3
-
1
4
)
=
1
2×3×4
=
3
32×4
=
1
3
3
8
1
3
(
1
4
-
1
5
)
=
1
4
4
15
验证:
1
3
(
1
4
-
1
5
)
=
1
3×4×5
=
4
42×5
=
1
4
4
15

(1)按照上述两个等式及其验证过程的基本思路,猜想
1
4
(
1
5
-
1
6
)
的变形结果并进行验证;
(2)针对上述各式反映的规律,写出用n(n为大于等于2的整数)表示的等式,并进行验证.
答案

(1)

1
4
(
1
5
-
1
6
)
=
1
5
5
24

验证:左边=

1
4×5×6
=
5
52×6
=
1
5
5
24
=右边,故正确;

(2)

1
n
(
1
n+1
-
1
n+2
)
=
1
n+1
n+1
n(n+2)

验证:左边=

n+1
n(n+1)(n+2)
=
n+1
n(n+1)2(n+2)
=
1
n+1
n+1
n(n+2)
=右边

故正确.

单项选择题
单项选择题