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joshnash
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µù¥U¤é: 2006-03-06
µoªí¼Æ: 16


 ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

¤p©úª¾¹D,¹ï©ó¥ô·N¥¿¾ã¼Æn,
¦³1+2+3...+n=n(n+1)/2
¥L©ó¬O²q´ú1^2+2^2+3^2+...+n^2=n(n+1)(n+2)/3
¦ý¬O,¤pµØ¤@¬Ý³o­Ó¦¡¤l´N¹ï¤p©ú»¡:
"¤£¹ï,
§A²qªº¤½¦¡¤¤,µ¥¸¹¥k¦¡¤¤ªº¤À¤lªº²Ä¤T­Ó¦]¬O¿ù¤F"

½Ð°Ý:
(1)¨Ì¾Ú¤pµØªº»¡ªk,¦p¦ó­×¥¿¤p©ú©Ò²q´úªº¤½¦¡
(¥²¶·µ¹¥X²z¥Ñ)
(2)§A¯à§_²q´ú1^3+2^3+3^3+...n^3ªº¨D©M¤½¦¡
½ÐÃÒ©ú

 2006-03-19 18:09­Ó¤H¸ê®Æ«ô³Xºô¯¸
ej0cl6
Home away from home



µù¥U¤é: 2005-03-03
µoªí¼Æ: 281
¡© ² ¡ª ªº ¬G ¶m

 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

1^2+2^2+3^2+4^2+...+n^2=n(n+1)(2n+1)/6
1^3+2^3+3^3+4^3+...+n^3=[n(n+1)/2]^2


_________________

http://blog.pixnet.net/ej0cl6
¡ô³o¬O³Ìªñ¦¨¥ßªº¼Æ¾ÇBLOG
¡@¸Ì­±¦³¨Ç´X¦óªºªF¦è
¡@¤j®a¥i¥H°ÑÆ[¬Ý¬Ý©O

 2006-03-19 21:24­Ó¤H¸ê®Æ«ô³Xºô¯¸
joshnash
Just popping in



µù¥U¤é: 2006-03-06
µoªí¼Æ: 16


 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

±z¯à§_¯d¤U­pºâ¹Lµ{@"@
¦nÅý¤p§Ì¤F¸Ñ
ÁÂÁ±z!

 2006-03-20 19:23­Ó¤H¸ê®Æ«ô³Xºô¯¸
ej0cl6
Home away from home



µù¥U¤é: 2005-03-03
µoªí¼Æ: 281
¡© ² ¡ª ªº ¬G ¶m

 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

¦]¬°n^2=n*n¤]¥i¥H»¡¬On­Ón¬Û¥[
©Ò¥H¥i±N£Un^2µe¦¨¤U¹Ï

£Un^2§Y¬°¹Ï¤¤©Ò¦³¼Æ¦rªºÁ`©M
±µ¤U¨Ó±ÛÂà¹Ïªº¤è¦V¦A¬Û¥[
+ + =
¦@¦³n(n+1)/2­Ó2n+1
§Y¥i±À¥X£Un^2=n(n+1)/2*(2n+1)/3=n(n+1)(2n+1)/6


_________________

http://blog.pixnet.net/ej0cl6
¡ô³o¬O³Ìªñ¦¨¥ßªº¼Æ¾ÇBLOG
¡@¸Ì­±¦³¨Ç´X¦óªºªF¦è
¡@¤j®a¥i¥H°ÑÆ[¬Ý¬Ý©O

 2006-03-20 20:17­Ó¤H¸ê®Æ«ô³Xºô¯¸
ej0cl6
Home away from home



µù¥U¤é: 2005-03-03
µoªí¼Æ: 281
¡© ² ¡ª ªº ¬G ¶m

 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

n^3«h¥i¥H³o¼Ë¬Ý
¡@¡@¡@¢°¡@¡@¢±¡@¡@¢²¡@¡@¢³¡@¡@¢´¡@......

¢°¡@¡@¢°¡@¡@¢±¡@¡@¢²¡@¡@¢³¡@¡@¢´¡@

¢±¡@¡@¢±¡@¡@¢³¡@¡@¢µ¡@¡@¢·¡@¡@¢°¢¯

¢²¡@¡@¢²¡@¡@¢µ¡@¡@¢¸¡@¡@¢°¢±¡@¢°¢´

¢³¡@¡@¢³¡@¡@¢·¡@¡@¢°¢±¡@¢°¢µ¡@¢±¢¯

¢´¡@¡@¢´¡@¡@¢°¢¯¡@¢°¢´¡@¢±¢¯¡@¢±¢´
......

«Ü®e©ö¥i¥H¬Ý¥X
1^3+2^3+3^3+4^3+..+n^3
´N¬O¹Ï¤¤¶Â¦â¼Æ¦rªºÁ`©M
¤]´N¬O
(1+2+3+4+...+n)^2=[n(n+1)/2]^2


_________________

http://blog.pixnet.net/ej0cl6
¡ô³o¬O³Ìªñ¦¨¥ßªº¼Æ¾ÇBLOG
¡@¸Ì­±¦³¨Ç´X¦óªºªF¦è
¡@¤j®a¥i¥H°ÑÆ[¬Ý¬Ý©O

 2006-03-20 20:24­Ó¤H¸ê®Æ«ô³Xºô¯¸
joshnash
Just popping in



µù¥U¤é: 2006-03-06
µoªí¼Æ: 16


 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

«D±`·PÁ±z¸Ô²Óªº¹Ï¸Ñ,
Åý¤p§Ì£¸¥ØÁAµM@@"
¯u¬O°ª©Û!

 2006-03-20 20:30­Ó¤H¸ê®Æ«ô³Xºô¯¸
frog
Just can't stay away



µù¥U¤é: 2006-02-28
µoªí¼Æ: 114


 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

¯uªº«Ü¨ØªA¦³¤H¥i¥H¥Î¹Ï¸Ñªk»PÆ[¹îªk¨Ó»¡©ú
¦ý¦b¥¿¦¡¦Ò¸Õ¤¤,¥H¹Ï¸Ñªk»PÆ[¹îªk¨ÓÃÒ©ú,©¹©¹¬O·|³Q¦©¤À¬Æ¦Ü¥þ¿ùªº
¥H¤U¥H"¼Æ¾ÇÂk¯Çªk"¨ÓÃÒ©ú,©|½Ð¤£§[«ü±Ð
1^2+2^2+3^2+4^2+...+n^2=n(n+1)(2n+1)/6
(1)·ín=1®É,­ì¦¡¦¨¥ß
(2)¥On=m®É­ì¦¡¦¨¥ß
§Y1^2+2^2+3^2+4^2+...+m^2=m(m+1)(2m+1)/6
«h·ín=m+1®É,¥N¤J­ì¦¡¥ª°¼:
1^2+2^2+3^2+4^2+...+m^2+(m+1)^2
=m(m+1)(2m+1)/6+(m+1)^2
=(m(m+1)(2m+1)+6(m+1)^2)/6
=(m+1)(m(2m+1)+6(m+1))/6
=(m+1)(2m^2+7m+6)/6
=(m+1)(m+2)(2m+3)/6
=(m+1)((m+1)+1)(2(m+1)+1)/6
¥Ñ¼Æ¾ÇÂk¯Çªk,¥»ÃD±oÃÒ
¦P²z¥ß¤è©M¤]¥i¥Î¼Æ¾ÇÂk¯ÇªkÃÒ©ú



 2006-03-20 22:33­Ó¤H¸ê®Æ
s87009812
Just can't stay away



µù¥U¤é: 2004-07-25
µoªí¼Æ: 132
¥x¥_¥«¥ß´º¬ü°ê¤¤

 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

³oÃD©|¥i§Q¥Î"®t¤Àªk"

£Un^2
¤½¦¡ n^3-(n-1)^3
n=1 1^3-0^3=3*1^2-3*1+1
n=2 2^3-1^3=3*2^2-3*2+1
n=3 3^3-2^3=3*3^2-3*3+1
.... .......................................
n=n n^3-(n-1)^3=3n^2-3*n+1

¬Û¥[±on^3-0^3=3=3£Un^2-3£Un^2-3£Un+1
3£Un^2=n^3+3£Un-n=n(n^2-1)+3n(n+1)/2=n(n+1)(2n+1)/2

¥t¥~·í£Un^3ªº®É­Ô,´N¥i¥H§Q¥Î
n^4-(n-1)^4,§Q¥Î¤W­±Ãþ¦üªº¸Ñªk,¾ã²zÂk¯Ç¥X¤½¦¡: n^2(n+1)^2/4


_________________
·Q­n±o¨ì¤°»ò,´N¥²¶·¥I¥X¬Û¦Pªº¥N»ù...³o´N¬O·Òª÷³N¤¤©Ò»¡ªº"µ¥»ù¥æ´«­ì«h".

 2006-03-20 22:48­Ó¤H¸ê®Æ¶Ç°e Email µ¹ s87009812
joshnash
Just popping in



µù¥U¤é: 2006-03-06
µoªí¼Æ: 16


 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

¤Þ¤å:

frog ¼g¹D:
¯uªº«Ü¨ØªA¦³¤H¥i¥H¥Î¹Ï¸Ñªk»PÆ[¹îªk¨Ó»¡©ú
¦ý¦b¥¿¦¡¦Ò¸Õ¤¤,¥H¹Ï¸Ñªk»PÆ[¹îªk¨ÓÃÒ©ú,©¹©¹¬O·|³Q¦©¤À¬Æ¦Ü¥þ¿ùªº
¥H¤U¥H"¼Æ¾ÇÂk¯Çªk"¨ÓÃÒ©ú,©|½Ð¤£§[«ü±Ð
1^2+2^2+3^2+4^2+...+n^2=n(n+1)(2n+1)/6
(1)·ín=1®É,­ì¦¡¦¨¥ß
(2)¥On=m®É­ì¦¡¦¨¥ß
§Y1^2+2^2+3^2+4^2+...+m^2=m(m+1)(2m+1)/6
«h·ín=m+1®É,¥N¤J­ì¦¡¥ª°¼:
1^2+2^2+3^2+4^2+...+m^2+(m+1)^2
=m(m+1)(2m+1)/6+(m+1)^2
=(m(m+1)(2m+1)+6(m+1)^2)/6
=(m+1)(m(2m+1)+6(m+1))/6
=(m+1)(2m^2+7m+6)/6
=(m+1)(m+2)(2m+3)/6
=(m+1)((m+1)+1)(2(m+1)+1)/6
¥Ñ¼Æ¾ÇÂk¯Çªk,¥»ÃD±oÃÒ
¦P²z¥ß¤è©M¤]¥i¥Î¼Æ¾ÇÂk¯ÇªkÃÒ©ú






°w¹ïfrog¥ý¥ÍªºÂk¯Çªk
§Ú¦³¤@ÂI¤p¤pªººÃ°Ý:
§AÁÙ¤£ª¾¹D
1^2+2^2+3^2+4^2+...+n^2=n(n+1)(2n+1)/6
µ¥¦¡ªº¥kÃ䪺¦¡¤l¬O¤°»ò®É
­n«ç»ò¥ÎÂk¯ÇªkÃÒ©ú?
±z³o¼Ë¤£¬O¥u¦³ÃÒ©ú¤F«e¤@¦ì¥ý¥ÍªºÃÒ©úµL»~¶Ü?
ÁÂÁÂ

 2006-03-23 19:49­Ó¤H¸ê®Æ«ô³Xºô¯¸
frog
Just can't stay away



µù¥U¤é: 2006-02-28
µoªí¼Æ: 114


 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

­n¨Ï¥Î¼Æ¾ÇÂk¯Çªk,¨ä«e´£¬O­n¦³¤@­Ó§¹¾ãªº¤èµ{¦¡©Îµ¥¦¡ ,©Ò¥H¥kÃ䪺¦¡¤ln(n+1)(2n+1)/6¬O¤£¥i©Î¯Êªº
¦¹ÃD¦]¬°¤ñ¸û²³æ,©Ò¥H¥i¥H¥Î¹Ï¸Ñªk©ÎÆ[¹îªkÂk¯Ç¥Xµ²ªG,¦ý·í§A¤W¤F°ª¤¤«á,§A´N·|µo²{«Ü¦hÃÒ©ú¬OµLªk¥HÆ[¹îªk±o¨ìªº,¦]¦¹§AÁÙ¬O­n¼ô±x¼Æ¾ÇÂk¯Çªk

 2006-03-23 20:32­Ó¤H¸ê®Æ
³X«È








 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

¯uªº«ÜªY½àej0cl6ªº¹Ï¸Ñªk
¥u¯à»¡¹ê¦b¬O¤Ó¼F®`¤F
³Ð³y¤Oº¡ÂI­C

 2006-03-24 02:37
j7631103
Home away from home



µù¥U¤é: 2005-03-06
µoªí¼Æ: 490


 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

§Úªº¦Ñ®v±Ðªk©M¥L¤@¼Ë

 2006-03-24 19:33­Ó¤H¸ê®Æ¶Ç°e Email µ¹ j7631103
bubupin
Home away from home



µù¥U¤é: 2007-03-13
µoªí¼Æ: 353


 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

¡@¡@¨Df(n)=1^k+2^k+3^k+4^k+.....n^k®É¤j®a¤ñ¸û¼ô±xªº¬Ok=1,2,3®Éªº¤½¦¡¸Ñ¡A¦¹®É¤]¦³¤@¨Ç´X¦ó¹Ï§ÎÀ°§U§Ú­Ì¤F¸Ñ¤½¦¡¦p¦ó¨D±o¡A¦ý§Ú»P¯«½Þ¤@°_ª±³o­Ó½ÒÃD®É¡A±o¨ì¤F¤@¨Ç¤ß±o¡A¦b¦¹´£¥X¨Ó¤À¨É¤]´M¨D«ü¥¿¡D

************************************************************
³]f(x)=1^k+2^k+3^k+4^k+.....x^k¡A¦]f(0)=0¡A¦h¶µ¦¡±`¼Æ¶µ¬°0¡A¬Gx|f(x)

·íx=n¡An|f(n)¡F·íx=n+1¡An+1|f(n+1)¡An+1|f(n)+(n+1)^k¡An+1|f(n)¡F¬Gn(n+1)|f(n)

·ín¬°°¸¼Æ¡Ax=2n+1®É¡A2n+1|f(2n+1)¡Ff(2n+1) (mod 2n+1)=2f(n) (mod 2n+1)=0¡A

¬G2n+1|f(n)¡A¦¹®Én(n+1)(2n+1)|f(n)

************************************************************

¥Ñ¤W¥iª¾¡A§Ú­Ì¥i³]A=n(n+1)¡A±N¦h¶µ¦¡f(n,k)=1^k+2^k+.....+n^k§ï³]¬°Aªº¨ç¼Æ

¨Ò¦p¡G¥Of(n,7)=A^2(A^2+aA+b)/8¡A¥Ñf(1,7)=1©Mf(2,7)=129¨D±oa=-4/3¡Ab=2/3¡F

¬Gf(n,7)=A^2(3A^2-4A+2)/24

************************************************************
¥OA=n(n+1)¡AB=n(n+1)(2n+1)/2¡A³]f(n,k)=1^k+2^k+......+n^k¡A¥Ñ¤W­z¤èªk¥i±o

f(n,1)=A/2¡Ff(n,2)=B/3¡Ff(n,3)=A^2/4¡Ff(n,4)= B(3A-1)/15=B(A-1/3)/5

f(n,5)=A^2*(2A-1)/12=A^2*(A-1/2)/6¡Ff(n,6)=B(3A^2-3A+1)/21=B(A^2-A+1/3)/7

f(n,7)=A^2(3A^2-4A+2)/24=A^2(A^2-4A/3+2/3)/8

f(n,8)=B(5A^3-10A^2+9A-3)/45=B(A^3-2A^2+9A/5-3/5)/9

f(n,9)=A^2(A-1)(2A^2-3A+3)/20= A^2(A-1)(A^2-3A/2+3/2)/10

************************************************************
¦ý¦¸¼Æ°ª¡A¼Æ¦rÅܤj®ÉÁÙ¬O·P¨ì¦³ÂI³Â·Ð¡A¦¹®É§Q¥Î¿n¤À­ì²z¨Ó°µ´N¤ñ¸û³æ¯Â¦h¤F¡D

±N³sÄò¯Å¼Æ©Mµø¬°¦h¶µ¦¡¤§¿n¤À¡A³]f(n,k)=n^k¡A«h¿n¤À¬°[1/(k+1)]*n^(k+1)+........

f(n,k)=[n^(k+1)+a*n^k+b*n^(k-1)+.......]/(k+1)

§Q¥Î¦¸¼Æ¬°k®É©Ò±of(n,k)¨Ó¨D±o¦¸¼Æk+1®É¤§f(n,k+1)

1.¦]¥²§tn(n+1)¦]¤l¡A¬G±`¼Æ¶µ¬°¹s

2.±Nf(n)­¼¥Hk«á©Ò±okf(n)»â¾É«Y¼Æ¬°1¡A¸g¿n¤À«á·s¼W¤@¦¸¶µ¡A³]«Y¼Æ¬°a¡A

§Î¦¨1/(k+1)*n^(k+1)+..........+a*n

¦p¦¹¨C¦¸¶È¥Îf(1)=1¡A¥O«Y¼Æ©M¬°1§Y¥i¸Ñ±oa¡A¨Ã²¤Æ¨Df(n)ªºµ{§Ç

 2007-10-23 22:41­Ó¤H¸ê®Æ
playdd1
Just popping in



µù¥U¤é: 2008-05-17
µoªí¼Æ: 1


 Re: ¼Æ¾Ç°ÝÃD.¨âÃDÃÒ©úÃD

³o¨Ç¸Ñªk¬O§Ú¦b®Ñ¥»¤W¬Ý¤£¨ìªº
­ì¥»·Q§â¤½¦¡­I¤@­I´N¦n
¦ý¬O¨}¤ß¤£¦w
¨S·Q¨ìÅý§Ú¹J¨ì³o»ò¦h°ª¤â
¯u¬O¶}¤ß
ÁöµM§ÚÁÙ¤£¬O«ÜÀ´
¦ý¤p¦³¶}¬
¤×¨ä¹ïn^2¬°¦ó§è¨ìn^3¥hÃÒ©ú¦³¤F¤@ÂI½u¯Á

¹Ï¸Ñªk¯u¬O·PÁÂ

 2008-05-17 11:56­Ó¤H¸ê®Æ


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Copyright 2000 ~2004¤E³¹¼Æ¾Ç¥Xª©ªÀ¡B¤E³¹¼Æ¾Ç°òª÷·|
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