Question

¹Û²ìÏÂÃæµÄËãʽ£¬²¢»Ø´ðÎÊÌ⣺

1

1¡Á2

=1-

1

2

£¬

1

2¡Á3

=

1

2

-

1

3

£¬

1

3¡Á4

=

1

3

-

1

4

£¬

1

4¡Á5

=

1

4

-

1

5

£¬¡­£¬°´´Ë¹æÂɼÆË㣺

1

1¡Á2

+

1

2¡Á3

=1

1

2

+

1

2

-

1

3

=

2

3

£¬

1

1¡Á2

+

1

2¡Á3

+

1

3¡Á4

=1-

1

2

+

1

2

-

1

3

+

1

3

-

1

4

=

3

4

£¬
¡­
£¨1£©¼ÆË㣺1-

1

1¡Á2

-

1

2¡Á3

-

1

3¡Á4

-¡­-

1

n(n+1)

£¨2£©

1

1¡Á3

+

1

3¡Á5

+

1

5¡Á7

+¡­£¬ËãʽÖÐÒѾ­Ð´³öÁË3¸ö·ÖÊý£¬Çëд³öµÚn¸ö·ÖÊý£®
£¨3£©¼ÆË㣺

1

1¡Á3

+

1

3¡Á5

+

1

5¡Á7

¡­+

1

99¡Á101

£®

Answer 1

¿¼µã£ºÓÐÀíÊýµÄ»ìºÏÔËËã

רÌ⣺¹æÂÉÐÍ

·ÖÎö£º£¨1£©Ô­Ê½ÀûÓõóöµÄ²ðÏî¹æÂɱäÐΣ¬¼ÆËã¼´¿ÉµÃµ½½á¹û£»
£¨2£©¹éÄÉ×ܽáµÃµ½¹æÂÉ£¬Ð´³öµÚn¸ö·ÖÊý¼´¿É£»
£¨3£©Ô­Ê½ÀûÓòðÏî¹æÂɱäÐΣ¬¼ÆËã¼´¿ÉµÃµ½½á¹û£®

½â´ð£º½â£º£¨1£©Ô­Ê½=1-1+

1

2

-

1

2

+

1

3

-¡­-

1

n

+

1

n+1

=

1

n+1

£»
£¨2£©µÚn¸ö·ÖÊýΪ

1

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

£»
£¨3£©Ô­Ê½=

1

2

£¨1-

1

3

+

1

3

-

1

5

+¡­+

1

99

-

1

101

£©=

1

2

¡Á

100

101

=

50

101

£®

µãÆÀ£º´ËÌ⿼²éÁËÓÐÀíÊýµÄ»ìºÏÔËË㣬ÊìÁ·ÕÆÎÕÔËËã·¨ÔòÊǽⱾÌâµÄ¹Ø¼ü£®