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ã©ããåºçºç¹ã«ããŠãã³ãŒã¹ã®æ°ã¯å€ãããªãããã®æ°ã a[n] éããšããã
以äžã§ã¯Aãåºçºç¹ãšããã³ãŒã¹ãèããã
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ãã®ã©ã¹ãåãBã®ã³ãŒã¹æ°(=ã©ã¹ãåãDã®ã³ãŒã¹æ°)ã b[n] éããšããã
n=0ã®å Žåãããªãã¡åãªãåè§åœ¢ABCDãèãããšãã©ã¹ãåãBã®ã³ãŒã¹ã¯
AâDâCâBâA
ã®1éããªã®ã§ã b[0]=1 ã§ããã a[0]=2 ã§ããã
nâ§1ã®å Žåã0段ç®ã®4ç¹ãéãé çªã¯ä»¥äžã®(1.1)ãã(3.4)ãŸã§ã®ããããã«ãªãã
(1.1) AâaââŠâbâBâCâDâA
(1.2) AâBâbââŠâcâCâDâA
(1.3) AâBâCâcââŠâdâDâA
(1.4) AâBâCâDâdââŠâaâA
(1.5) AâaââŠâdâDâCâBâA
(1.6) AâDâdââŠâcâCâBâA
(1.7) AâDâCâcââŠâbâBâA
(1.8) AâDâCâBâbââŠâaâA
(2.1) AâaââŠâbâBâCâcââŠâdâDâA
(2.2) AâBâbââŠâcâCâDâdââŠâaâA
(2.3) AâaââŠâdâDâCâcââŠâbâBâA
(2.4) AâDâdââŠâcâCâBâbââŠâaâA
(3.1) AâaââŠâcâCâBâbââŠâdâDâA
(3.2) AâBâbââŠâdâDâCâcââŠâaâA
(3.3) AâaââŠâcâCâDâdââŠâbâBâA
(3.4) AâDâdââŠâbâBâCâcââŠâaâA
ã(1.1)ïœ(1.8)ã®å Žå
(1.1)ãäŸã«èãããšãéšååaââŠâbã®ç®æã¯n-1åã®ç«æ¹äœã§Aãåºçºãã©ã¹ãåãBã«ãªãã³ãŒã¹æ°ã«çããã®ã§ b[n-1] éããšãªãã
(1.2)ïœ(1.8)ãåæ§ã« b[n-1] éãã§ããã
ã(2.1)ïœ(2.4)ã®å Žå
(2.1)ãäŸã«èããã
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šäœãšããŠãã¹ãŠã®ç¹ãéãã®ã§ãi,jã®å°ãªããšãäžæ¹ã¯nãšãªãã
i=j=nã®å Žåãäºã€ã®éšååã¯ã©ã¡ãããå³ãžçŽé²ãn段ç®ã§ã²ãšã€é£ãžç§»åãå·ŠãžçŽé²ããŠæ»ãã³ãŒã¹ãããªãã®ã§ã1éãã§ããã
i<j=nã®å ŽåãéšååaââŠâbã¯å³ãžçŽé²ãi段ç®ã§ã²ãšã€é£ãžç§»åãå·ŠãžçŽé²ããŠæ»ãããšã«ãªãã
éšååcââŠâdã¯i+1段ç®ãŸã§å³ãžçŽé²ãi+1段ç®ä»¥éã®ãã¹ãŠã®ç¹ãéã£ãåŸi+1段ç®ãã1段ç®ãŸã§å·ŠãžçŽé²ããããšã«ãªãã
ãã£ãŠãã®å Žåã¯ãn-i-1åã®ç«æ¹äœã§Aãåºçºãã©ã¹ãåãBã«ãªãã³ãŒã¹æ°ã«çããã®ã§ b[n-i-1] éããšãªãã
j<i=nã®å Žåãåæ§ã« b[n-i-1] éããšãªãã
以äžã®3ã€ã®å Žåã®æ°ãåããããš(2.1)ã®ã³ãŒã¹æ°ã¯ã
1+2*Σ[i=1..n-1]b[n-i-1]
= 1+2*Σ[k=0..n-2]b[k]
éããšãªãã
(2.2)ïœ(2.4)ãåæ§ã« 1+2*Σ[k=0..n-2]b[k] éãã§ããã
ã(3.1)ïœ(3.4)ã®å Žå
çµè«ãšããŠããã®å Žåã®ã³ãŒã¹ã¯ååšããªãããã®çç±ã以äžã«ç€ºãã
(3.1)ãäŸã«èããã
éšååaââŠâcã®æé«å°é段ãi段ç®ãéšååbââŠâdã®æé«å°é段ãj段ç®ãšããã
ã³ãŒã¹ã¯å
šäœãšããŠãã¹ãŠã®ç¹ãéãã®ã§ãi,jã®å°ãªããšãäžæ¹ã¯nãšãªãã
i=j=nã®å Žåãn段ç®ãŸã§ã¯çŽé²ã§è¡ãæ¥ãããããªãããn段ç®ã§ã®äºã€ã®éšååã®ã³ãŒã¹ã®äž¡ç«ãã§ããªãããããã¯ããããªãã
i<j=nã®å ŽåãéšååaââŠâcã¯i段ç®ãŸã§ã®ã©ããã®æ®µã§ãã®æ®µå
ã®å°ãªããšã3ç¹ãéãããšã«ãªããã
ãããããšéšååbââŠâdããã®æ®µãåŸåŸ©ã§ééããããã®å°ãªããšã2ç¹ã確ä¿ã§ããªãããããããªãã
j<i=nã®å Žåãåæ§ã§ããã
以äžããã(3.1)ã®ã³ãŒã¹ã¯ååšããªãããšããããã
(3.2)ïœ(3.4)ãåæ§ã§ããã
ãããŸã§ã®ããšããnâ§1ã®ãšã次ã®äºã€ã®åŒãåŸãããã
a[n] = 8*b[n-1] + 4*(1+2*Σ[k=0..n-2]b[k]) âŠåŒâ
b[n] = 3*b[n-1] + 1*(1+2*Σ[k=0..n-2]b[k]) âŠåŒâ¡
åŒâ¡ããb[n]ã®æŒžååŒ
b[n] = 1 + 3*b[n-1] + 2*Σ[k=0..n-2]b[k] âŠåŒâ¡'
ãåŸãããa[n]ã¯åŒâ ãã
a[n] = 4 + 8*Σ[k=0..n-1]b[k] âŠåŒâ '
ã䜿ãã°æ±ãŸãã
GAIããã®åé¡[B]ã¯n=4ã®å Žåãªã®ã§ãèšç®ãããš a[4]=612 éããšãªãã
ã©ãã§ãããããGAIãããæ±ããæ°ãšåãã«ãªããŸããã§ããããïŒ
No.1262ããã²ã2023幎7æ7æ¥ 20:15
[A]
(1) 2918(éã)
(2) 2188(éã)
(3) 2116(éã)
ã§ãã£ãã®ã«å¯Ÿã
[B]
ã®å
ã«æ»ããã³ãŒã¹ã«éå®ãããšãã©ã®é ç¹ããåºçºãããã
å
ã«æ»ããã³ãŒã¹æ°ã¯äžå®ã§ãããã²ããããæ±ããããŠãã612(éã)
ãããŸããã
[A]ã®å Žåã®æ§ã«åºçºç¹ãç°ãªãã°åœç¶ç°ãªãçµæãèµ·ããã ãããšèšç®ãããããŠã¿ããšã
åæã«æ±ºãã€ããŠããããšãèŠäºã«è£åãããŸããã
ãããåŸããèããŠã¿ãããéããçµè·¯ã¯ããããžãŒçã«ã©ããåããã®ã§ããããšã«ãªãããã«æããŠçŽåŸããŸããã
é ã®äžã ãã§ãã®612ãèŠã€ããããããšã«é©ããŸããã
No.1263GAI2023幎7æ8æ¥ 08:34
ã©ããããã£ãŠããããã§ããã£ãã§ãã
n=4ã§æãç«ã£ãŠããã®ãªãã°æŒžååŒã¯æ£ããå¯èœæ§ãé«ãã§ããã
ãšãããŸã§æžããåŸããµãšæãç«ã£ãŠèª¿ã¹ãŠã¿ããã
OEISã®A003699ã«ããã«ãã³éè·¯ã®æ°(=ã³ãŒã¹æ°ã®åå)ãèŒã£ãŠããŸããã
ããã®ïŒé
é挞ååŒãèŠãã«ãã£ãšã·ã³ãã«ãªèãæ¹ãããããã§ãã
æåããååã®æ°ã§æ€çŽ¢ããŠããã°ããã£ãã®ãâŠâŠã
ç§ã®æçš¿ã§äžãæééããŠããã®ã§ä¿®æ£ããŸãã
ã(2.1)ïœ(2.4)ã®å Žåãã®äž
誀ïŒãj<i=nã®å Žåãåæ§ã« b[n-i-1] éããšãªããã
â
æ£ïŒãj<i=nã®å Žåãåæ§ã«èã㊠b[n-j-1] éããšãªããã
No.1264ããã²ã2023幎7æ8æ¥ 13:11
ããã²ãããã®ãšèãæ¹ãå
±éããéšåãå€ãã§ããããããªã®ã§ã©ãã§ãããã
n åã®ç«æ¹äœãå·Šå³äžåã«äžŠã¹ãŠããå Žåã§èããŸãã
æãå³åŽã«ããç«æ¹äœã®å³åŽã®é¢ã®é ç¹ 4 ã€ã¯ã
ã»4 ã€ãé£ç¶ããŠéãïŒã³åïŒ
ã»2 ã€ããŸãéããåŸã§æ®ã 2 ã€ãéãïŒäºåïŒ
ã®ããããã§éãããšã«ãªããŸãã
ã³åã®ãã¿ãŒã³æ°ã ã³[n], äºåã®ãã¿ãŒã³æ°ã äº[n] ãšããŸãã
ã³åã«ã€ããŠãå³åŽã®é¢ã® 4 é ç¹ãåé€ããŠçµè·¯ãç絡ããããšãèããŸãã
GAI ããã®å³ã§äŸãæããã°ãâŠâŠâ4â5â6â16â15â14ââŠâŠ ã âŠâŠâ4â14ââŠâŠ ã«ç絡ãããããªã€ã¡ãŒãžã§ãã
ç«æ¹äœ n+1 åã®å Žåã®ã³åã®çµè·¯å
šãŠãç絡ãããšã
ç«æ¹äœ n åã®å Žåã®ã³åã®çµè·¯å
šçš®ã 3 ã€ãã€ããã³äºåã®çµè·¯å
šçš®ã 2 ã€ãã€ã§ããã®ã§ã
ã³[n+1] = 3*ã³[n] + 2*äº[n]
äºåã«ã€ããŠãåæ§ã«èããŸãã
GAI ããã®å³ã§äŸãæããã°ãâŠâŠâ4â5â6â7ââŠâŠâ17â16â15â14ââŠâŠ ã âŠâŠâ4â7ââŠâŠâ17â14ââŠâŠ ã«ç絡ãããããªã€ã¡ãŒãžã§ãã
ç«æ¹äœ n+1 åã®å Žåã®äºåã®çµè·¯å
šãŠãç絡ãããšã
ç«æ¹äœ n åã®å Žåã®ã³åã®çµè·¯å
šçš®ã 1 ã€ãã€ããã³äºåã®çµè·¯å
šçš®ã 1 ã€ãã€ã§ããã®ã§ã
äº[n+1] = ã³[n] + äº[n]
䞡挞ååŒãã ã³[n] ãæ¶å»ããŠ
äº[n+2] = 4*äº[n+1] - äº[n]
ãŸããã³[1] = 8, äº[1] = 4 ãªã®ã§ãäº[2] = 12, äº[3] = 44, äº[4] = 164, äº[5] = 612
ãã£ãŠãæ±ããç·æ°ã¯ ã³[4] + äº[4] = äº[5] = 612 éãã§ãã
ããã«ãã³éè·¯æ°ããa[n] = (1/2)*ã³[n] + (1/2)*äº[n] = (1/2)*äº[n+1] ãšèãããšã
a[1] = 6, a[2] = 22, a[n+2] = 4*a[n+1] - a[n]
ãšãã挞ååŒãæãç«ã€ããšã瀺ãããŸãã
ïŒããã§ã¯ n ãç«æ¹äœæ°ãšããŠèããŠããã®ã§ãA003699 ãšã¯ n ã®å€ã 1 ã€ãããŸãïŒ
No.1290DD++2023幎7æ11æ¥ 10:17
DD++æ§ãããã«ã¡ã¯ã
ãäºé
å®çã®äžæè°ããã
ïŒïŒïŒïŒïŒïŒïŒïŒ
(4)åŒããã
ãã(1+1)^n=1^n +nC1 1^(n-1) +nC2 1^(n-2)+nC3 1^(n-3)+ã»ã»ã»ã»+nC(n-1) 1+1
ãã(2+1)^n=2^n +nC1 2^(n-1) +nC2 2^(n-2)+nC3 2^(n-3)+ã»ã»ã»ã»+nC(n-1) 2+1
ãã(3+1)^n=3^n +nC1 3^(n-1) +nC2 3^(n-2)+nC3 3^(n-3)+ã»ã»ã»ã»+nC(n-1) 3+1
ãã(4+1)^n=4^n +nC1 4^(n-1) +nC2 4^(n-2)+nC3 4^(n-3)+ã»ã»ã»ã»+nC(n-1) 4+1
ãã(5+1)^n=5^n +nC1 5^(n-1) +nC2 5^(n-2)+nC3 5^(n-3)+ã»ã»ã»ã»+nC(n-1) 5+1
ããããããããããããããããããããã»
ãã(r+1)^n=r^n +nC1 r^(n-1) +nC2 r^(n-2)+nC3 r^(n-3)+ã»ã»ã»ã»+nC(n-1) r+1
ããããããããããããããããããããã»
+)ã(a+1)^n=a^n +nC1 a^(n-1) +nC2 a^(n-2)+nC3 a^(n-3)+ã»ã»ã»ã»+nC(n-1) a+1
ã¯ã
ãã(1+1)^n-1^n= nC1 1^(n-1) +nC2 1^(n-2)+nC3 1^(n-3)+ã»ã»ã»ã»+nC(n-1) 1+1
ãã(2+1)^n-2^n= nC1 2^(n-1) +nC2 2^(n-2)+nC3 2^(n-3)+ã»ã»ã»ã»+nC(n-1) 2+1
ãã(3+1)^n-3^n= nC1 3^(n-1) +nC2 3^(n-2)+nC3 3^(n-3)+ã»ã»ã»ã»+nC(n-1) 3+1
ãã(4+1)^n-4^n= nC1 4^(n-1) +nC2 4^(n-2)+nC3 4^(n-3)+ã»ã»ã»ã»+nC(n-1) 4+1
ãã(5+1)^n-5^n= nC1 5^(n-1) +nC2 5^(n-2)+nC3 5^(n-3)+ã»ã»ã»ã»+nC(n-1) 5+1
ããããããããããããããããããããã»
ãã(r+1)^n-r^n= nC1 r^(n-1) +nC2 r^(n-2)+nC3 r^(n-3)+ã»ã»ã»ã»+nC(n-1) r+1
ããããããããããããããããããããã»
+)ã(a+1)^n-a^n= nC1 a^(n-1) +nC2 a^(n-2)+nC3 a^(n-3)+ã»ã»ã»ã»+nC(n-1) a+1
---------------------------------------------------------------------------------
ãã(a+1)^n-1^n=nC1{ã»ã»â ã»ã»}+nC2{ã»ã»â¡ã»ã»}+nC3{ã»ã»â¢ã»ã»}+ã»ã»ã»ã»+nC(n-1){ã»ã»(n-1)ã»ã»}+a
ãã(a+1)^n=nC1{ã»ã»â ã»ã»}+nC2{ã»ã»â¡ã»ã»}+nC3{ã»ã»â¢ã»ã»}+ã»ã»ã»ã»+nC(n-1){ã»ã»(n-1)ã»ã»}+a+1
ãšããã§ãnãçŽ æ°ãªãã°nCsã¯åžžã«nã®åæ°ã§ãããã
(a+1)^n=nB+(a+1)---(6)
ãã ããnB=nC1{ã»ã»â ã»ã»}+nC2{ã»ã»â¡ã»ã»}+nC3{ã»ã»â¢ã»ã»}+ã»ã»ã»ã»+nC(n-1){ã»ã»(n-1)ã»ã»}
ïŒïŒïŒïŒïŒïŒ
ããã§ã
ãã(a+1)^n=nC1{ã»ã»â ã»ã»}+nC2{ã»ã»â¡ã»ã»}+nC3{ã»ã»â¢ã»ã»}+ã»ã»ã»ã»+nC(n-1){ã»ã»(n-1)ã»ã»}+a+1
ãã(a+1)^n-(a+1)=nC1{ã»ã»â ã»ã»}+nC2{ã»ã»â¡ã»ã»}+nC3{ã»ã»â¢ã»ã»}+ã»ã»ã»ã»+nC(n-1){ã»ã»(n-1)ã»ã»}
nãåææ°ãªããnCsã¯åžžã«nã®åæ°ã«ãªããªããã(ãã ã0<s<n)
(a+1)^n-(a+1)â nB---(6)'
ãããã£ãŠã
nãçŽ æ°ã®ãšãã ã
(a+1)^n=nB+(a+1)
(a+1)^n-(a+1)=nB
α^n-α=nBãïŒãã ãα=a+1ã)
ãæãç«ã€ã
ç·šéæžã¿
No.1265ããããã¯ã¡ã¹ã2023幎7æ8æ¥ 16:54
ãn ãçŽ æ°ã®ãšãã«æç«ãããã¯æ£ããã§ããã
ããããããã¯ãn ãåææ°ã®ãšãã¯äžæç«ã§ããããã©ããã«ã¯çŽæ¥é¢ä¿ããªãããã䞻匵ããããªãå¥é蚌æãå¿
èŠã§ãã
No.1266DD++2023幎7æ8æ¥ 21:16
åäŸããããšããŸããã
561 = 3*11*17 ã¯åææ°ã§ãã
N^561 - N = ( N^3 - N ) * ( N^558 + N^556 + âŠâŠ + 1 )
ã«ãããŠã3 ã¯çŽ æ°ãªã®ã§ N^3 - N 㯠3 ã®åæ°ãN^558 + N^556 + âŠâŠ + 1 ã¯æŽæ°ã§ãã
ãããã£ãŠãN^561 - N 㯠3 ã®åæ°ã§ãã
N^561 - N = ( N^11 - N ) * ( N^550 + N^540 + âŠâŠ + 1 )
ã«ãããŠã11 ã¯çŽ æ°ãªã®ã§ N^11 - N 㯠11 ã®åæ°ãN^550 + N^540 + âŠâŠ + 1 ã¯æŽæ°ã§ãã
ãããã£ãŠãN^561 - N 㯠11 ã®åæ°ã§ãã
N^561 - N = ( N^17 - N ) * ( N^544 + N^527 + âŠâŠ + 1 )
ã«ãããŠã17 ã¯çŽ æ°ãªã®ã§ N^17 - N 㯠17 ã®åæ°ãN^544 + N^527 + âŠâŠ + 1 ã¯æŽæ°ã§ãã
ãããã£ãŠãN^561 - N 㯠17 ã®åæ°ã§ãã
以äžãããN^561 - N 㯠561 ã®åæ°ã§ãã
No.1267DD++2023幎7æ8æ¥ 23:42
DD++æ§ããã¯ããããããŸãã
ããããã«N^33-NãèŠãŠã¿ãŸãããã33=3*11ã§ããã
(%i1) factor(N^33-N);åŒã®å æ°å解ãã
(%o1)(N - 1) N (N + 1) (N^2 + 1) (N^4 + 1) (N^8 + 1) (N^16 + 1)
33ãN^11ã¯ãããŸããããå æ°å解ã¯äžéãããã§ããŸããã®ã§ããã以å€ãªãã¯ãã§ãã
ãææã®åäŸã¯äžé©åœã ãšæããŸãã
No.1268ããããã¯ã¡ã¹ã2023幎7æ9æ¥ 07:23
(%i2) factor(N^561-N);åŒã®å æ°å解ãã
(%o2) (N - 1) N (N + 1) (N^2 + 1) (N^4 + 1) (N4 - N^3 + N^2 - N + 1)
(N^4 + N^3 + N^2 + N + 1) (N^6 - N^5 + N^4 - N^3 + N^2 - N + 1)
(N^6 + N^5 + N^4 + N^3 + N^2 + N + 1) (N^8 + 1) (N^8 - N^6 + N^4 - N^2 + 1)
(N^12 - N^10 + N^8 - N^6 + N^4 - N^2 + 1) (N^16 - N^12 + N^8 - N^4 + 1)
(N^24 - N^20 + N^16 - N^8 + N^4 - N + 1) (N^24 - N^23 + N^19 - N^18 + N^17 - N^16 + N^14 - N^13 + N^12 - N^11 + N^10 - N^8+ N^7- N^6+ N^5 - N + 1)
(N^24 + N^23 - N^19 - N^18 - N^17 - N^16 + N^14 + N^13 + N^12 + N^11 + N^10- N^8- N^7 - N^6- N^5+ N + 1) (N^32 - N^24 + N^16 - N^8 + 1)
(N^48 - N^40 + N^32 - N^24 + N^16 - N^8 + 1)
(N^48 + N^46 - N^38 - N^36 - N^34 - N^32 + N^28 + N^26 + N^24 + N^22 + N^20- N^16 - N^14 - N^12 - N^10 + N^2 + 1)
(N^96 + N^92 - N^76 - N^72 - N^68 - N^64 + N^56 + N^52 + N^48 + N^44 + N^40 -N^32 - N^28 - N^24 - N^20 + N^4 + 1)
(N^192+ N^184- N^152- N^144- N^136- N^128+ N^112 + N^104+ N^96 + N^88 + N^80- N^64 - N^56 - N^48 - N^40 + N^8 + 1)
ãšãªãã561ããããŸãããã3,11,17ããããŸããã
No.1269ããããã¯ã¡ã¹ã2023幎7æ9æ¥ 07:48
ç§ã¯ã561 ã®å Žåã«äŸå€çãªçŸè±¡ãçºçããããšèšã£ãŠããã®ã«ãªãç¡é¢ä¿ãª 33 ã®è©±ãå§ããã®ã§ããïŒ
ç§ã¯ 33 ãäŸå€ã ãªããŠäžèšãèšã£ãŠããŸãããã
No.1270DD++2023幎7æ9æ¥ 07:53
ïŒnãåææ°ãªããnCsã¯sã«ãããããåžžã«nã®åæ°ã«ãªããªããã(ãã ã0<s<n)
(a+1)^n-(a+1)â nB
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No.1271ããããã¯ã¡ã¹ã2023幎7æ9æ¥ 08:17
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誀ã1ïŒ
ãnãåææ°ãªããnCsã¯sã«ãããããåžžã«nã®åæ°ã«ãªããªããã¯åœã§ãã
åäŸã¯ã6C1 = 6 ã 9C4 = 126 ãªã©ãããã§ãã
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n ã®åæ°ã§ãªãæ°ã®åèšã n ã®åæ°ã§ãªãæ°ã«ãªãä¿èšŒã¯ãããŸããã
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No.1272DD++2023幎7æ9æ¥ 08:50
nCsã§ãs=1ãããs=n-1ãŸã§ããã¹ãŠïœã®åæ°ã§ãªããšãå³èŸºã¯nã§ããããŸããã
6C1ã¯ããããããããŸãããïŒC2ãïŒC3,ïŒC4,ïŒC5ã¯ã©ãã§ããïŒ
6C1=2x3
6C2=3x5
6C3=2^2x5
6C4=3x5
6C5=2x3
>(a+1)^n-(a+1)=nC1{ã»ã»â ã»ã»}+nC2{ã»ã»â¡ã»ã»}+nC3{ã»ã»â¢ã»ã»}+ã»ã»ã»ã»+nC(n-1){ã»ã»(n-1)ã»ã»}
å³èŸºã¯ãn=6ã§ããããªãã§ãããïŒ
No.1273ããããã¯ã¡ã¹ã2023幎7æ9æ¥ 09:11
3+5 㯠4 ã§ããããªããŠã 4 ã®åæ°ã§ãããšãã話ãããŠããã®ã§ããã
No.1274DD++2023幎7æ9æ¥ 10:14
ïŒn ã®åæ°ã§ãªãæ°ã®åèšã n ã®åæ°ã§ãªãæ°ã«ãªãä¿èšŒã¯ãããŸããã
åäŸã¯ãn=4 ã«å¯Ÿãã4 ã®åæ°ã§ãªã 3 ãš 5 ã®å㯠4 ã®åæ°ã§ãã
(a+1)^n-(a+1)=nAã®è©±ã§ãnåææ°ã®å Žåã
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ã«ããããŠãnCsãã¿ãªnã®åæ°ã«ãªããªãã®ã§ãå³èŸºã¯ïœã§ããããªããšãªãããã§ããããã®è©±ãšã©ãã«ãããããããã®ã§ããããïŒ
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No.1275ããããã¯ã¡ã¹ã2023幎7æ9æ¥ 10:53
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No.1276å£ããæ2023幎7æ9æ¥ 11:14
å£ããææ§ãããã«ã¡ã¯ã
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No.1277ããããã¯ã¡ã¹ã2023幎7æ9æ¥ 11:37
ïŒïœã§ããããªããŠãå³èŸºãïœã®åæ°ã«ãªãå¯èœæ§ã¯ãããŸãããã
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N^p-N=çŽæ°ã®å
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No.1278ããããã¯ã¡ã¹ã2023幎7æ9æ¥ 11:50
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No.1279DD++2023幎7æ9æ¥ 11:54
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No.1280ããããã¯ã¡ã¹ã2023幎7æ9æ¥ 12:00
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No.1281DD++2023幎7æ9æ¥ 12:03
AïŒBãªãã°Cã§ããããïœãçŽ æ°ãªãã°ãa^n-a=ïœAããªããã€ã
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No.1282ããããã¯ã¡ã¹ã2023幎7æ9æ¥ 12:23
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å°ãèœã¡çããŠããé¡ãããŸãã
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No.1283Dengan kesaktian Indukmu2023幎7æ9æ¥ 16:18
ãAïŒB ãªãã°Cã§ãããã®åŠå®ã¯ããAïŒB ã〠ã§ãªãããªã®ã§ã
ãïœ ãçŽ æ°ãªãã°ãa^n-a=ïœAãã®åŠå®ã¯ããïœ ãçŽ æ° ã〠a^n-aâ ïœAãã§ãããïŒ
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No.1284HP管çè
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No.1285ããããã¯ã¡ã¹ã2023幎7æ9æ¥ 18:25
ïŒåã®ã·ã³ãã«ãªåœ¢ã®è§£ãèŠã€ãããŸããã
ïŒ2a-b,a+3b,3a+2b,-3a-2b,-a-3b,-2a+b)
ããšã(a-2b,2a+3b,3a+b,-3a-b,-2a-3b,-a+2b)
ããã«ãïŒåã«ææŠäžã§ãã
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No.1261ks2023幎7æ6æ¥ 19:46
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No.1229ks2023幎6æ28æ¥ 09:46
ksæ§ãããã«ã¡ã¯ã
ïŒä»»æã®èªç¶æ°Nã«ãå¶æ°ã足ããŠããã°ãå¹³æ¹æ°ã«ãªãããšãåãããŸããã
ããã¯ãN+(1+2+3+ã»ã»+N-1)x2=N+N(N-1)=N+N^2-N=N^2
ã§ããã
No.1230ããããã¯ã¡ã¹ã2023幎6æ28æ¥ 14:09
N+(2+4ïŒâŠïŒïŒïŒN-1ïŒïŒïŒNã®2ä¹
ãæãç«ã¡ã
NïŒïŒ6ïŒâŠïŒïŒNïŒN-1ïŒïŒ= âä¿®æ£ããŸããã
NïŒïŒ3ã®åæ°ã®æ°åã®åïŒïŒNã®ïŒä¹
NïŒïŒïŒã®åæ°ã®æ°åã®åïŒïŒNã®ïŒä¹ ãäžæç«
NïŒïŒïŒã®åæ°ã®æ°åã®åïŒïŒNã®ïŒä¹ãæãç«ã¡ãŸã
No.1232ks2023幎6æ29æ¥ 18:45
ïŒNïŒN-1ïŒãã
ïŒã®ïŒä¹ïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒïŒ
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No.1235ks2023幎6æ30æ¥ 19:10
ksæ§ããã¯ããããããŸãã
N+a1+a2+ã»ã»+aNã«ãããŠã
aN=3N(N-1)
ãšãããšã
a1=0
a2=6
a3=18
a4=36
a5=60
ã»ã»ã»ã»
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ãN
N+â{3i(i-1)}
ãi=1
ããN
=N+3â{i(i-1)}
ããi=1
ããNããããN
=N+3âi^2ãŒ3âi
ããi=1ãããi=1
=N+3N(N+1)(2N+1)/6ãŒ3N(N+1)/2
=N+(3/2)N(N+1){(2N+1)/3ãŒ1}
=N+(3/2)N(N+1){(2N+1-3)/3)}
=N+(3/2)N(N+1){(2N-2)/3)}
=N+N(N+1)(N-1)=N+N(N^2-1)
=N+N^3-N=N^3
ãããã£ãŠã
N+a1+a2+a3ã»ã»+aN=N^3
N+0+6+18+ã»ã»+3N(N-1)=N^3
N+6+18+ã»ã»+3N(N-1)=N^3
No.1236ããããã¯ã¡ã¹ã2023幎7æ1æ¥ 09:17
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No.1237ããããã¯ã¡ã¹ã2023幎7æ1æ¥ 09:35
N+x1+x2+ã»ã»+xN=N^4ã«ãããŠã
xN=aN^3+bN^2+cN+dãšãããšã
N+â(ai^3+bi^2+ci+d)
=N+a{N^2(N+1)^2}/4+b{N(N+1)(2N+1)}/6+c{N(N+1)}/2+dN
=a{N^2(N+1)^2}/4+b{N(N+1)(2N+1)}/6+c{N(N+1)}/2+(d+1)N
N^4={3aN^4+(6a+4b)N^3+(3a+6b+6c)N^2+(2b+6c+12d+12)N}/12
ãã£ãŠã
3a/12=1 a/4=1 ããã«a=4
6a+4b=0 3a+2b=0 12+2b=0ããã«b=-6
3a+6b+6c=0 a+2b+2c=0 4-12+2c=0ããã«c=4
2b+6c+12d+12=0ãb+3c+6d+6=0 -6+12+6d+6=0 12+6d=0ããã«d=-2
ãã£ãŠã
xN=aN^3+bN^2+cN+d
=2(N-1)(2N^2-N+1)
ãããã£ãŠã
x1=0
x2=14
x3=64
x4=174
ã»ã»ã»ã»
ãã£ãŠã
N+x1+x2+ã»ã»+xN=N^4
N+0+14+64+174+ã»ã»ã»+2(N-1)(2N^2-N+1)=N^4
ããŠã
N=1 ã®ãšãïŒ
N=2 ã®ãšã2+0+14=16=2^4
N=3 ã®ãšã3+0+14+64=81=3^4
N=4 ã®ãšã3+0+14+64+174=256=4^4
No.1240ããããã¯ã¡ã¹ã2023幎7æ1æ¥ 18:24
N^4åã®å
¬åŒ{N(N+1)(2N+1)(3N^2+3N-1}/30ããã
N+x1+x2+ã»ã»+xN=N^5ã«ãããŠã
xN=aN^4+bN^3+cN^2+dN+eãšãããšã
ãN
N+â(ai^4+bi^3+ci^2+di+e)
ãi=1
=N+a{N(N+1)(2N+1)(3N^2+3N-1}/30+b{N^2(N+1)^2}/4+c{N(N+1)(2N+1)}/6+d{N(N+1)}/2+eN
=a{N(N+1)(2N+1)(3N^2+3N-1}/30+b{N^2(N+1)^2}/4+c{N(N+1)(2N+1)}/6+d{N(N+1)}/2+(e+1)N
N^5={12aN^5+(30a+15b)N^4+(20a+30b+20c)N^3+(15b+30c+30d)N^2+(-2a+10c+30d+60e+60)}/60
ãã£ãŠã
12a/60=1 a/5=1 ããã«a=5
30a+15b=0 2a+b=0 10+b=0ããã«b=-10
20a+30b+20c=0 2a+3b+2c=0 10-30+2c=0ããã«c=10
15b+30c+30d=0ãb+2c+2d=0 -10+20+2d=0 10+2d=0ããã«d=-5
-2a+10c+30d+60e+60=0ã -a+5c+15d+30e+30=0ã -5+50-75+30e+30=0ã 0+30e=0
ããã«e=0
ãã£ãŠã
xN=aN^4+bN^3+cN^2+dN+e
=5N^4-10N^3+10N^2-5N
=5N(N-1)(N^2-N+1)
ãããã£ãŠã
x1=0
x2=30
x3=210
x4=780
ã»ã»ã»ã»
ãã£ãŠã
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N+0+30+210+780+ã»ã»ã»+5N(N-1)(N^2-N+1)=N^5
ããŠã
N=1ã®ãšãã1+0=1
N=2ã®ãšãã2+0+30=32=2^5
N=3ã®ãšãã3+0+30+210=243=3^5
N=4ã®ãšãã4+0+30+210+780=1024=4^5
No.1241ããããã¯ã¡ã¹ã2023幎7æ1æ¥ 19:03
N^5åã®å
¬åŒ{N^2(N+1)^2(2N^2+2N-1)}/12ããã
N+x1+x2+ã»ã»+xN=N^6ã«ãããŠã
xN=aN^5+bN^4+cN^3+dN^2+eN+fãšãããšã
ãN
N+â(ai^5+bi^4+ci^3+di^2+ei+f)
ãi=1
=N+a{N^2(N+1)^2(2N^2+2N-1)}/12+b{N(N+1)(2N+1)(3N^2+3N-1)}/30+c{N^2(N+1)^2}/4+d{N(N+1)(2N+1)}/6+e{N(N+1)}/2+fN
=a{N^2(N+1)^2(2N^2+2N-1)}/12+b{N(N+1)(2N+1)(3N^2+3N-1)}/30+c{N^2(N+1)^2}/4+d{N(N+1)(2N+1)}/6+e{N(N+1)}/2+(f+1)N
N^6={10aN^6+(30a+12b)N^5+(25a+30b+15c)N^4+(20b+30c+20d)N^3+(-5a+15c+30d+30e)N^2+(-2b+10d+30e+60f+60)N}/60
ãã£ãŠã
10a/60=1 a/6=1 ããã«a=6
30a+12b=0 5a+2b=0 30+2b=0ããã«b=-15
25a+30b+15c=0 5a+6b+3c=0 30-90+3c=0 -60+3c=0ããã«c=20
20b+30c+20d=0ã2b+3c+2d=0 -30+60+2d=0 30+2d=0ããã«d=-15
-5a+15c+30d+30e=0 -a+3c+6d+6e=0 -6+60-90+6e=0 -36+6e=0ããã«e=6
-2b+10d+30e+60f+60=0 30-150+180+60f+60=0 120+60f=0 12+6f=0ããã«f=-2
ãã£ãŠã
xN=aN^5+bN^4+cN^3+dN^2+eN+f
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=(N-1)(6N^4-9N^3+11N^2-4N+2)
ãããã£ãŠã
x1=0
x2=62
x3=664
x4=3366
ã»ã»ã»ã»
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N+x1+x2+ã»ã»+xN=N^6
N+0+62+664+3366+ã»ã»ã»+(N-1)(6N^4-9N^3+11N^2-4N+2)=N^6
ããŠã
N=1ã®ãšãã1+0=1
N=2ã®ãšãã2+0+62=64=2^6
N=3ã®ãšãã3+0+62+664=729=3^6
N=4ã®ãšãã4+0+64+664+3366=4096=4^6
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¬åŒ{N(N+1)(2N+1)(3N^4+6N^3-3N+1)}/42ããã
N+x1+x2+ã»ã»+xN=N^7ã«ãããŠã
xN=aN^6+bN^5+cN^4+dN^3+eN^2+fN+gãšãããšã
ãN
N+â(ai^6+bi^5+ci^4+di^3+ei^2+fi+g)
ãi=1
=N+a{N(N+1)(2N+1)(3N^4+6N^3-3N+1)}/42+b{N^2(N+1)^2(2N^2+2N-1)}/12+c{N(N+1)(2N+1)(3N^2+3N-1)}/30+d{N^2(N+1)^2}/4+e{N(N+1)(2N+1)}/6+f{N(N+1)}/2+gN
=a{N(N+1)(2N+1)(3N^4+6N^3-3N+1)}/42+b{N^2(N+1)^2(2N^2+2N-1)}/12+c{N(N+1)(2N+1)(3N^2+3N-1)}/30+d{N^2(N+1)^2}/4+e{N(N+1)(2N+1)}/6+f{N(N+1)}/2+(g+1)N
N^7={60aN^7+(210a+70b)N^6+(210a+210b+84c)N^5+(175b+210c+105d)N^4+(-70a+140c+210d+140e)N^3+(-35b+105d+210e+210f)N^2+(10a-14c+70e+210f+420g+420)N}/420
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210a+70b=0 3a+b=0 21+b=0ããã«b=-21
210a+210b+84c=0 5a+5b+2c=0 35-105+2c=0 -70+2c=0ããã«c=35
175b+210c+105d=0ã5b+6c+3d=0 -105+210+3d=0 105+3d=0ããã«d=-35
-70a+140c+210d+140e=0 -a+2c+3d+2e=0 -7+70-105+2e=0 -42+2e=0ããã«e=21
-35b+105d+210e+210f=0 -b+3d+6e+6f=0 21-105+126+6f=0 42+6f=0 ããã«f=-7
10a-14c+70e+210f+420g+420=0 5a-7c+35e+105f+210g+210=0 210g=0 ããã«g=0
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xN=aN^6+bN^5+cN^4+dN^3+eN^2+fN+g
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=7N(N-1)(N^2-N+1)^2
ãããã£ãŠã
x1=0
x2=126
x3=2058
x4=14196
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N+x1+x2+ã»ã»+xN=N^7
N+0+126+2058+14196+ã»ã»ã»+7N(N-1)(N^2-N+1)^2=N^7
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N=1ã®ãšãã1+0=1
N=2ã®ãšãã2+0+126=128=2^7
N=3ã®ãšãã3+0+126+2058=2187=3^7
N=4ã®ãšãã4+0+126+2058+14196=16384=4^7
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