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#16
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![]() 12 + 123 + 1234 + ... + 123456789 + 1234567891 + 12345678912 + 123456789123
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠![]() |
#17
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![]() ⨷Âì¹èÒʹ㨠(áÅÐÂÒ¡) ÁÒ¡¤ÃѺ ¾Í¨ÐºÍ¡·ÕèÁÒä´éäËÁ¤ÃѺ
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#18
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![]() ![]()
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠![]() 01 ÁԶعÒ¹ 2006 20:36 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Mastermander |
#19
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![]() ¼Á¤Ò´ÇèҤӵͺÍÒ¨à»ç¹ $$ \Big \lfloor \frac{ 13717421 \cdot 10^{n+2} }{999999999} \Big \rfloor -5 \Big \lfloor \frac{n}{9} \Big \rfloor -f(n \bmod 9)$$ â´Â·Õè $$f(0)=1, f(1)=1, f(2)=2, f(3)=2, f(4)=3$$ $$f(5)=4, f(6)=5, f(7)=6, f(8)=6$$ áµèÂѧäÁèä´é¾ÔÊÙ¨¹ì¤ÃѺ ¤Ô´ÇèÒ¹èÒ¨ÐÂÒ¡ÁÒ¡æ áÅéÇ¡çÂѧäÁèá¹èã¨ÇèÒ simplify µèÍä´éÃÖà»ÅèÒ Âѧ䧢ʹÙà©ÅÂ˹èÍÂä´éäËÁ¤ÃѺ
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#20
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![]() $$ \frac{1}{81}\bigg[\frac{100(10^{n+1}-1)}{9}-\frac{9n^2+47n+200}{2}\bigg] $$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠![]() |
#21
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![]() Êٵ÷Õè¹éͧ Mastermander ãËéÁÒ¹Ñè¹à»ç¹¼ÅºÇ¡ÂèÍ¢ͧ͹ءÃÁ
12 + 123 + 1234 + 12345 + 123456 + 1234567 + 12345678 + 123456789 + 1234567900 + 12345679020 + 123456790230 + 1234567902340 + 12345679023450 + 123456790234560 + ... äÁèãªè¢Í§Íѹ·Õèà»ç¹¤Ó¶ÒÁ¤ÃѺ |
#22
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![]() ÍèÒÇ â·É·Õ¤ÃѺ ¼Áà¢éÒ㨼Դ ã¹Ë¹Ñ§Ê×ÍãËéÁÒá¤è¹Ñé¹
ÁÕ⨷ÂìÁÒ½Ò¡ 1 ¢éͤÃѺ ¨§ËÒ¤èҢͧ͹ءÃÁ͹ѹµì $$\frac{1}{6}+\frac{1}{36}+\frac{3}{216}+\frac{17}{1296}+\frac{83}{7776}+...$$ $\frac14$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠![]() 26 ÁÕ¹Ò¤Á 2007 00:40 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ nongtum à˵ؼÅ: triple posts merged |
#23
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![]() ·ÓäÁ¼ÁÃÙéÊÖ¡ÇèÒà·ÍÁáá¢Í§Í¹Ø¡ÃÁ Áѹ´ÙäÁèà¢éҾǡ¡Ñºà·ÍÁ·ÕèàËÅ×ÍàÅÂÅèÐà¹Õè (ËÃ×ÍÇèÒàÃÒà¢éÒ㨼Դàͧ
![]() ÂÑ§ä§ ¶éÒäÁèÅÓºÒ¡ÁÒ¡¹Ñ¡ ú¡Ç¹¹éͧ Mastermander à¢Õ¹à·ÍÁµèÍä»(à·ÍÁ·Õè 6) ã¹Í¹Ø¡ÃÁ ãËé´Ù«Ñ¡¹Ô´¹Ö§ä´éäËÁ¤ÃѺ à¾ÃÒе͹¹Õé¼Á¤Ô´ä´é 2/7 «Öè§äÁèµÃ§¡Ñºà©ÅÂ
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ |
#24
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![]() next term
$$ \frac{345}{46656} $$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠![]() |
#25
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![]() ÊÃØ»ÇèÒ à·ÍÁÅèÒÊØ´·Õè¹éͧ Mastermander post äÇé äÁèµÃ§¡Ñº·Õè¼Á¤Ô´ áÊ´§ÇèÒ¼Á¤§à¢éÒ㨼Դàͧ
![]() §Ñé¹¼ÁÇèÒ¹éͧ Mastermander ŧ $ a_n $äÇéãËé¹Ô´¹Ö§¡ç´Õ¤ÃѺ ¶×Íà»ç¹¡ÒÃà©ÅÂä»ã¹µÑÇ (äÁèµéͧáÊ´§ÇÔ¸Õ·Ó¡çä´é¤ÃѺ) áÅШҡ⨷Âì¢é͹Õé ¡ç·ÓãËé¼Áä´éäÍà´Õ à¡Ô´¤Ó¶ÒÁãËÁè¢Öé¹ÁÒ Evaluate $$ \frac{1}{6}+\frac{3}{36}+\frac{17}{216}+\frac{83}{1296}+\frac{417}{7776} +\cdots $$ $\frac{5}{7}$
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ 01 ÁԶعÒ¹ 2006 20:37 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ passer-by |
#26
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![]() $$ a_n=\frac{3^n-n(2^n)}{6^n} $$
ÍéÒ§ÍÔ§:
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠![]() |
#27
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![]() á¡éä¢ãËéàÃÕºÃéÍÂáÅéǤÃѺ
¨Ò¡¤Ó¶ÒÁ¢Í§¤Ø³ Mastermander ·ÓãËé¼ÁÃÙéÊÖ¡ä´é·Ñ¹·ÕÇèÒ ¡ÒõéͧÁÒ¤ÅÓËÒ sequence àͧ¹ÕèÁѹà˹×èͨÃÔ§æ Êèǹ¢éͧ͢¼Á áÁé $ a_n $ ẺÊÓàÃç¨ÃÙ» ¨ÐäÁèÊÇÂà·èÒ¢éÍ¡è͹˹éÒ áµèÇÔ¸Õ·Õè¨Ð¹Óä»ÊÙè $a_n $ ãËé¼ÅºÒ§ÍÂèÒ§·ÕèÊÇÂãªéä´éàŤÃѺ
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ |
#28
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![]() $\displaystyle{ \frac{1}{6}+\frac{3}{36}+\frac{17}{216}+\frac{83}{1296}+\frac{417}{7776} +\cdots }$
$= \displaystyle{ \frac{1}{6}+\frac{5(1)-2}{36}+\frac{5(3)+2}{216}+\frac{5(17)-2}{1296}+\frac{5(83)+2}{7776} +\cdots } $
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠![]() |
#29
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![]() ¡ÒáÃШÒÂà·ÍÁ ¢Í§¹éͧ Mastermander ãªéä´éàŤÃѺ
¨ÃÔ§æ ¨Ò¡µÑÇàÈÉ ¨ÐàËç¹ä´éàÅÂÇèÒ Áѹà»ç¹ recurrence relation $ a_1=1 \quad a_2=3 $ $ a_{n}=5a_{n-2}+4a_{n-1} \quad n\geq 3 $
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ |
#30
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![]() áÅéǨФԴ¼ÅÃÇÁ(¤ÓµÍº) ä´éÍÂèÒ§ääÃѺ
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