´óÊýѧ¼Ò¸ß˹ÔÚÉÏѧ¶ÁÊéʱÔø¾­Ñо¿¹ýÕâÑùÒ»¸öÎÊÌ⣺1+2+3+¡­+n=£¿¾­¹ýÑо¿£¬Õâ¸öÎÊÌâµÄ½áÂÛÊÇ1+2+3+¡­+n=
1
2
n(n+1)
£¬ÆäÖÐnÊÇÕýÕûÊý£®ÏÖÔÚÎÒÃÇÀ´Ñо¿Ò»¸öÀàËƵÄÎÊÌ⣺1¡Á2+2¡Á3+¡­+n£¨n+1£©=£¿¹Û²ìÏÂÃæÈý¸öÌØÊâµÄµÈʽ£º
1¡Á2=
1
3
(1¡Á2¡Á3-0¡Á1¡Á2)
£¬
2¡Á3=
1
3
(2¡Á3¡Á4-1¡Á2¡Á3)
£¬
3¡Á4=
1
3
(3¡Á4¡Á5-2¡Á3¡Á4)
£¬
½«ÕâÈý¸öµÈʽµÄÁ½±ßÏà¼Ó£¬¿ÉÒԵõ½1¡Á2+2¡Á3+3¡Á4=
1
3
¡Á3¡Á4¡Á5=20
£®
¸ù¾ÝÉÏÊö¹æÂÉ£¬ÇëÄã¼ÆË㣺1¡Á2+2¡Á3+¡­+n£¨n+1£©=
1
3
n(n+1)(n+2)
1
3
n(n+1)(n+2)
£»1¡Á2¡Á3+2¡Á3¡Á4+¡­+n£¨n+1£©£¨n+2£©=
1
4
n(n+1)(n+2)(n+3)
1
4
n(n+1)(n+2)(n+3)
£®
·ÖÎö£º¹Û²ìÒÑÖªµÄÈý¸öµÈʽ£¬µÃ³öÒ»°ãÐԵĹæÂÉ£¬¸ù¾ÝµÃ³öµÄ¹æÂɱíʾ³ö1¡Á2+2¡Á3+¡­+n£¨n+1£©µÄÿһÏµÖÏûºÏ²¢ºó¼´¿ÉµÃµ½½á¹û£»ÒÀ´ËÀàÍƵõ½1¡Á2¡Á3=
1
4
£¨1¡Á2¡Á3¡Á4-0¡Á1¡Á2¡Á3£©£¬2¡Á3¡Á4=
1
4
£¨2¡Á3¡Á4¡Á5-1¡Á2¡Á3¡Á4£©£¬
×ܽá³öÒ»°ãÐÔ¹æÂÉ£¬½«¸÷Ïî±äÐκó£¬È¥À¨ºÅºÏ²¢¼´¿ÉµÃµ½½á¹û£®
½â´ð£º½â£º¸ù¾ÝÔĶÁ²ÄÁÏÖеÄÀý×ӵãº1¡Á2+2¡Á3+¡­+n£¨n+1£©
=
1
3
£¨1¡Á2¡Á3-0¡Á1¡Á2£©+
1
3
£¨2¡Á3¡Á4-1¡Á2¡Á3£©+¡­+
1
3
[n£¨n+1£©£¨n+2£©-£¨n-1£©n£¨n+1£©]
=
1
3
n£¨n+1£©£¨n+2£©£»
ÒÀ´ËÀàÍÆ£º1¡Á2¡Á3=
1
4
£¨1¡Á2¡Á3¡Á4-0¡Á1¡Á2¡Á3£©£¬2¡Á3¡Á4=
1
4
£¨2¡Á3¡Á4¡Á5-1¡Á2¡Á3¡Á4£©£¬
¡à1¡Á2¡Á3+2¡Á3¡Á4+¡­+n£¨n+1£©£¨n+2£©
=
1
4
£¨1¡Á2¡Á3¡Á4-0¡Á1¡Á2¡Á3£©+
1
4
£¨2¡Á3¡Á4¡Á5-1¡Á2¡Á3¡Á4£©+¡­+
1
4
[£¨n£¨n+1£©£¨n+2£©£¨n+3£©-£¨n-1£©n£¨n+1£©£¨n+2£©]=
1
4
n£¨n+1£©£¨n+2£©£¨n+3£©£®
¹Ê´ð°¸Îª£º
1
3
n£¨n+1£©£¨n+2£©£»
1
4
n£¨n+1£©£¨n+2£©£¨n+3£©
µãÆÀ£º´ËÌ⿼²éÁ˹æÂÉÐÍ£ºÊý×ֵı仯À࣬ÆäÖÐŪÇåÌâÒ⣬µÃ³öÒ»°ãÐԵĹæÂÉÊǽⱾÌâµÄ¹Ø¼ü£®
Á·Ï°²áϵÁдð°¸
Ïà¹ØÏ°Ìâ

¿ÆÄ¿£º³õÖÐÊýѧ À´Ô´£º ÌâÐÍ£º

´óÊýѧ¼Ò¸ß˹ÔÚÉÏѧ¶ÁÊéʱÔø¾­Ñо¿¹ýÕâÑùÒ»¸öÎÊÌ⣺1+2+3¡­+100=£¿£¬¾­¹ýÑо¿£¬Õâ¸öÎÊÌâµÄÒ»°ãÐÔ½áÂÛÊÇ1+2+3¡­+n=
1
2
n(n+1)
£¬ÆäÖÐnÊÇÕýÕûÊý£¬ÏÖÔÚÎÒÃÇÀ´Ñо¿Ò»¸öÀàËƵÄÎÊÌ⣺1¡Á2+2¡Á3+¡­+n£¨n+1£©=£¿
¹Û²ìÏÂÃæÈý¸öÌØÊâµÄµÈʽ£º
1¡Á2=
1
3
(1¡Á2¡Á3-0¡Á1¡Á2)

2¡Á3=
1
3
£¨2¡Á3¡Á4-1¡Á2¡Á3£©
3¡Á4=
1
3
(3¡Á4¡Á5-2¡Á3¡Á4)

½«ÕâÈý¸öµÈʽµÄÁ½±ßÏà¼Ó£¬¿ÉÒԵõ½1¡Á2+2¡Á3+3¡Á4=
1
3
¡Á3¡Á4¡Á5=20
¶ÁÍêÕâ¶Î²ÄÁÏ£¬Çë³¢ÊÔÇó£¨ÒªÇóд³ö¹æÂÉ£©£º
£¨1£©1¡Á2+2¡Á3+3¡Á4+4¡Á5=£¿
£¨2£©1¡Á2+2¡Á3+¡­+100¡Á101=£¿
£¨3£©1¡Á2+2¡Á3+¡­+n£¨n+1£©=£¿

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º³õÖÐÊýѧ À´Ô´£º ÌâÐÍ£ºÔĶÁÀí½â

ÔĶÁ²ÄÁÏ£¬´óÊýѧ¼Ò¸ß˹ÔÚÉÏѧ¶ÁÊéʱÔø¾­Ñо¿¹ýÕâÑùÒ»¸öÎÊÌ⣺1+2+3+¡­+100=£¿¾­¹ýÑо¿£¬Õâ¸öÎÊÌâµÄÒ»°ãÐÔ½áÂÛÊÇ1+2+3+4+5+¡­+n=
1
2
n(n+1)
£¬ÆäÖÐnÊÇÕýÕûÊý£®ÏÖÔÚÎÒÃÇÀ´Ñо¿Ò»¸öÀàËƵÄÎÊÌ⣺
¹Û²ìÏÂÃæÈý¸öÌØÊâµÄµÈʽ£º
1¡Á2+2¡Á3+3¡Á4+¡­+n£¨n+1£©=
1¡Á2=
1
3
£¨1¡Á2¡Á3-0¡Á1¡Á2£©
2¡Á3=
1
3
£¨2¡Á3¡Á4-1¡Á2¡Á3£©
3¡Á4=
1
3
£¨3¡Á4¡Á5-2¡Á3¡Á4£©
½«ÕâÈý¸öµÈʽµÄÁ½±ß·Ö±ðÏà¼Ó£¬¿ÉÒԵõ½1¡Á+2¡Á3+3¡Á4=
1
3
¡Á3¡Á4¡Á5=20
¶ÁÍêÕâ¶Î²ÄÁÏ£¬ÇëÄã˼¿¼ºó»Ø´ð£º
£¨1£©1¡Á2+2¡Á3+3¡Á4+¡­+100¡Á101=
 
£®
£¨2£©1¡Á2+2¡Á3+3¡Á4+¡­+n£¨n+1£©=
 
£®
£¨3£©1¡Á2¡Á3+2¡Á3¡Á4+¡­+n£¨n+1£©£¨n+2£©=
 
£®
£¨Ö»Ðèд³ö½á¹û£¬²»±ØдÖмäµÄ¹ý³Ì£©

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º³õÖÐÊýѧ À´Ô´£º ÌâÐÍ£ºÔĶÁÀí½â

ÔĶÁ²ÄÁÏ£¬´óÊýѧ¼Ò¸ß˹ÔÚÉÏѧ¶ÁÊéʱÔø¾­Ñо¿¹ýÕâÑùÒ»¸öÎÊÌ⣺1+2+3+¡­+100=£¿¾­¹ýÑо¿£¬Õâ¸öÎÊÌâµÄÒ»°ãÐÔ½áÂÛÊÇ1+2+3+¡­+n=
1
2
n(n+1)
£¬ÆäÖÐnÊÇÕýÕûÊý£®ÏÖÔÚÎÒÃÇÀ´Ñо¿Ò»¸öÀàËƵÄÎÊÌ⣺1¡Á2+2¡Á3+¡­n£¨n+1£©=£¿
¹Û²ìÏÂÃæÈý¸öÌØÊâµÄµÈʽ1¡Á2=
1
3
(1¡Á2¡Á3-0¡Á1¡Á2)
£¬2¡Á3=
1
3
(2¡Á3¡Á4-1¡Á2¡Á3)
£¬3¡Á4=
1
3
(3¡Á4¡Á5-2¡Á3¡Á4)

¶ÁÍêÕâ¶Î²ÄÁÏ£¬ÇëÄã˼¿¼ºó»Ø´ð£º
£¨1£©5¡Á6=
 
=
 

½«Ç°ÃæÁ½¸öµÈʽµÄÁ½±ßÏà¼Ó£¬¿ÉÒԵõ½
1¡Á2+2¡Á3=
1
3
¡Á2¡Á3¡Á4=8
½«ÕâÈý¸öµÈʽµÄÁ½±ßÏà¼Ó£¬¿ÉÒԵõ½
1¡Á2+2¡Á3+3¡Á4=
1
3
¡Á3¡Á4¡Á5=20

¶ÁÍêÕâ¶Î²ÄÁÏ£¬ÇëÄã˼¿¼ºó»Ø´ð£º
£¨2£©1¡Á2+2¡Á3+¡­+100¡Á101=
 
=
 

£¨3£©1¡Á2+2¡Á3+¡­+n£¨n+1£©=
 
=
 
£®

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º³õÖÐÊýѧ À´Ô´£º ÌâÐÍ£ºÔĶÁÀí½â

ÔĶÁ²ÄÁÏ£º´óÊýѧ¼Ò¸ß˹ÔÚÉÏѧ¶ÁÊéʱÔø¾­Ñо¿¹ýÕâÑùÒ»¸öÎÊÌ⣺1+2+3+¡­+100=£¿¾­¹ýÑо¿£¬Õâ¸öÎÊÌâµÄÒ»°ãÐÔ½áÂÛÊÇ1+2+3+¡­+n=
1
2
n£¨n+1£©£¬ÆäÖÐnÊÇÕýÕûÊý£®ÏÖÔÚÎÒÃÇÀ´Ñо¿Ò»¸öÀàËƵÄÎÊÌ⣺1¡Á2+2¡Á3+¡­n£¨n+1£©=£¿
¹Û²ìÏÂÃæÈý¸öÌØÊâµÄµÈʽ£º
1¡Á2=
1
3
£¨1¡Á2¡Á3-0¡Á1¡Á2£©£¬
2¡Á3=
1
3
£¨2¡Á3¡Á4-1¡Á2¡Á3£©£¬
3¡Á4=
1
3
£¨3¡Á4¡Á5-2¡Á3¡Á4£©£¬
½«ÕâÈý¸öµÈʽµÄÁ½±ßÏà¼Ó£¬¿ÉÒԵõ½£º
1¡Á2+2¡Á3+3¡Á4=
1
3
(1¡Á2¡Á3-0¡Á1¡Á2+2¡Á3¡Á4-1¡Á2¡Á3
+3¡Á4¡Á5-2¡Á3¡Á4£©
=
1
3
¡Á3¡Á4¡Á5
=20
¶ÁÍêÕâ¶Î²ÄÁÏ£¬ÇëÄã˼¿¼ºó»Ø´ð£º
£¨1£©1¡Á2+2¡Á3+¡­+7¡Á8=
168
168
£»
£¨2£©1¡Á2+2¡Á3+¡­+n£¨n+1£©=
1
3
n£¨n+1£©£¨n+2£©
1
3
n£¨n+1£©£¨n+2£©
£»
£¨3£©Èô1¡Á2+2¡Á3+¡­+n£¨n+1£©=
1
3
¡Á9¡Á10¡Á11
£¬Çón±ßÐεÄÄڽǺͶÈÊý£®

²é¿´´ð°¸ºÍ½âÎö>>

¿ÆÄ¿£º³õÖÐÊýѧ À´Ô´£º ÌâÐÍ£ºÔĶÁÀí½â

ÔĶÁ²ÄÁÏ£¬´óÊýѧ¼Ò¸ß˹ÔÚÉÏѧ¶ÁÊéʱÔø¾­Ñо¿¹ýÕâÑùÒ»¸öÎÊÌ⣺1+2+3+¡­+100=£¿ÎÒÃÇ¿ÉÒÔÏÈ´Ó¼òµ¥µÄ¼¸¸öÊý¿ªÊ¼£¬¼ÆËã¡¢¹Û²ì£¬Ñ°Çó¹æÂÉ£¬µÃ³öÒ»°ãÐԵĽáÂÛ£®1=
1¡Á2
2
=1
£¬1+2=
2¡Á3
2
=3£¬1+2+3=
3¡Á4
2
=6£¬1+2+3+4=
4¡Á5
2
=10
£»¡­£¬
£¨1£©¼ÆË㣺1+2+3+¡­+100=
5050
5050
£®
£¨2£©¼ÆË㣺41+42+43+¡­+100=
5050
5050
-
820
820
=
4230
4230
£®

²é¿´´ð°¸ºÍ½âÎö>>

ͬ²½Á·Ï°²á´ð°¸