问题 填空题
求:
1
2
1+
1
2
+
1
3
(1+
1
2
)×(1+
1
3
)
+
1
4
(1+
1
2
)×(1+
1
3
)×(1+
1
4
)
+…+
1
99
(1+
1
2
)×(1+
1
3
)×…×(1+
1
99
)
答案

1
2
1+
1
2
+
1
3
(1+
1
2
)×(1+
1
3
)
+
1
4
(1+
1
2
)×(1+
1
3
)×(1+
1
4
)
+…+
1
99
(1+
1
2
)×(1+
1
3
)×…×(1+
1
99
)

=

1
2
3
2
+
1
3
3
2
×
4
3
+
1
4
3
2
×
4
3
×
5
4
+…+
1
99
3
2
×
4
3
×…×
100
99

=

2
2×3
+
2
3×4
+
2
4×5
+…+
2
99×100

=2×(

1
2×3
+
1
3×4
+
1
4×5
+…+
1
99×100
),

=2×(

1
2
-
1
3
+
1
3
-
1
4
+
1
4
-
1
5
+…+
1
99
-
1
100
),

=2×(

1
2
-
1
100
),

=

49
50

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