Question

S＝22+122?1+32+132?1+42+142?1+…+n2+1n2?1+(n+1)2+1(n+1)2?1

S＝22+122?1+32+132?1+42+142?1+…+n2+1n2?1+(n+1)2+1(n+1)2?1．

Answer 1

原式=（1+

2

22?1

）+（1+

2

32?1

）+（1+

2

42?1

）+…+（1+

2

(n+1)2?1

）
=n+

2

1×3

+

2

2×4

+

2

3×5

+…+

2

n(n+2)

=n+1-

1

3

+

1

2

-

1

4

+

1

3

-

1

5

+…+

1

n

-

1

n+2

=n+1+

1

2

-

1

n+1

-

1

n+2

=n+

3

2

-

1

n+1

-

1

n+2

．