求和：1/1×4+1/4×7+…+1/(3n-2)×(3n+1)= _ ．

题目

求和：

1

1×4

+

1

4×7

+…+

1

(3n-2)×(3n+1)

答案

设S_n=

1

1×4

+

1

4×7

+…+

1

(3n-2)×(3n+1)

则3S_n=

3

1×4

+

3

4×7

+…+

3

(3n-2)×(3n+1)

=1-

1

4

+

1

4

-

1

7

+…+

1

(3n-2)

-

1

(3n+1)

=1-

1

(3n+1)

=

3n

(3n+1)

所以S_n=

n

(3n+1)

．
故答案为

n

(3n+1)