问题 选择题
1
22-1
+
1
32-1
+
1
42-1
+…+
1
(n+1)2-1
的值为(  )
A.
n+1
2(n+2)
B.
3
4
-
n+1
2(n+2)
C.
3
4
-
1
2
(
1
n+1
+
1
n+2
)
D.
3
2
-
1
n+1
-
1
n+2
答案

1
22-1
+
1
32-1
+
1
42-1
+…+
1
(n+1)2-1

=

1
(2+1)(2-1)
+
1
(3+1)(3-1)
+
1
(4+1)(4-1)
+…+
1
(n+1+1)(n+1-1)

=

1
3×1
+
1
4×2
+
1
5×3
+…+
1
(n+2)n

=

1
2
[(1-
1
3
)+(
1
2
-
1
4
)  +(
1
3
-
1
5
)+…+ (
1
n
-
1
n+2
)
+(
1
2
-
1
4
)
+(
1
3
-
1
5
)
+…+(
1
n
-
1
n+2
)
]

=

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

=

3
4
-
1
2
(
1
n+1
+
1
n+2
).

故选C.

问答题
单项选择题