(1/n)=(1/a1)+(1/a2)ãã®é¢ä¿åŒãæºããçµåãã®èª¿æ»
2=>
[3, 6]
*1/2=1/3+1/6ã®åŒãæãç«ã€ããšã瀺ãã
3=>
[4, 12]
4=>
[6, 12]
5=>
[6, 30]
6=>
[10, 15]
9=>
[12, 36]
10=>
[15, 30]
--------------------------
(1/n)=(1/a1)+(1/a2)+(1/a3)
2=>
[4, 6, 12]
3=>
[6, 10, 15]
4=>
[10, 12, 15]
5=>
[12, 15, 20]
6=>
[12, 21, 28]
7=>
[15, 21, 35]
9=>
[20, 30, 36]
[21, 28, 36]
10=>
[21, 35, 42]
------------------------------------
(1/n)=(1/a1)+(1/a2)+(1/a3)+(1/a4)
2=>
[4, 10, 12, 15]
3=>
[9, 10, 15, 18]
4=>
[9, 18, 21, 28]
[10, 15, 21, 28]
5=>
[15, 20, 21, 28]
6=>
[20, 21, 28, 30]
7=>
[18, 28, 36, 42]
[20, 28, 30, 42]
9=>
[20, 35, 60, 63]
10=>
[30, 36, 45, 60]
-----------------------------------------------
(1/n)=(1/a1)+(1/a2)+(1/a3)+(1/a4)+(1/a5)
2=>
[6, 9, 10, 15, 18]
3=>
[10, 12, 15, 21, 28]
4=>
[12, 20, 21, 28, 30]
5=>
[18, 21, 28, 30, 36]
6=>
[21, 28, 30, 36, 45]
7=>
[28, 30, 36, 42, 45]
9=>
[35, 36, 45, 60, 63]
10=>
[28, 45, 63, 70, 84]
[30, 42, 60, 70, 84]
[30, 45, 60, 63, 84]
[36, 42, 45, 70, 84]
--------------------------------------------
(1/n)=(1/a1)+(1/a2)+(1/a3)+(1/a4)+(1/a5)+(1/a6)
2=>
[5, 9, 18, 20, 21, 28]
[6, 9, 12, 18, 21, 28]
[6, 10, 12, 15, 21, 28]
[7, 9, 12, 14, 18, 28]
[7, 10, 12, 14, 15, 28]
3=>
[10, 15, 20, 21, 28, 30]
4=>
[18, 20, 21, 28, 30, 36]
5=>
[20, 21, 35, 36, 42, 45]
6=>
[21, 35, 36, 42, 45, 60]
7=>
[28, 35, 42, 45, 60, 63]
9=>
[35, 42, 60, 63, 70, 84]
10=>
[42, 45, 60, 70, 84, 90]
----------------------------------------------------
(1/n)=(1/a1)+(1/a2)+(1/a3)+(1/a4)+(1/a5)+(1/a6)+(1/a7)
2=>
[9, 10, 12, 15, 18, 21, 28]
3=>
[14, 15, 20, 21, 28, 30, 35]
4=>
[15, 21, 30, 35, 36, 42, 45]
5=>
[21, 30, 35, 36, 42, 45, 60]
6=>
[28, 30, 35, 45, 60, 63, 70]
[30, 35, 36, 42, 45, 60, 70]
7=>
[30, 35, 45, 60, 63, 70, 84]
9=>
[42, 45, 60, 63, 84, 90, 105]
10=>
[42, 60, 63, 70, 84, 105, 126]
[45, 60, 63, 70, 84, 90, 126]
ïŒä»ã«ãå€ãã®é¢ä¿åŒãååšã§ããŸããæåŸã«çŸããæ°ããªãã ãå°ãããªã
ãã®ãéžãã§æ²ç€ºããŠããŸãã
-----------------------------------------------------------
å¹³æ¹æ°ã§ã®é¢ä¿åŒã§ã¯
(1/n)^2=(1/a1)^2+(1/a2)^2+(1/a3)^2ãã®é¢ä¿åŒãæºããçµåãã®èª¿æ»
6=>
[7, 14, 21]
*(1/6)^2=(1/7)^2+(1/14)^2+(1/21)^2 ãæç«ããããšã瀺ãã
----------------------------------------------------------
(1/n)^2=(1/a1)^2+(1/a2)^2+(1/a3)^2+(1/a4)^2
4=>
[5, 7, 28, 35]
----------------------------------------------------------
(1/n)^2=(1/a1)^2+(1/a2)^2+(1/a3)^2+(1/a4)^2+(1/a5)^2
4=>
[6, 7, 12, 14, 21]
6=>
[7, 15, 21, 42, 105]
9=>
[12, 14, 60, 252, 420]
10=>
[12, 21, 36, 252, 1260]
-----------------------------------------------------------
(1/n)^2=(1/a1)^2+(1/a2)^2+(1/a3)^2+(1/a4)^2+(1/a5)^2+(1/a6)^2
3=>
[4, 6, 7, 60, 84, 420]
4=>
[6, 7, 14, 15, 20, 21]
5=>
[6, 10, 30, 35, 70, 105]
6=>
[7, 12, 60, 105, 140, 420]
[7, 15, 20, 60, 84, 420]
7=>
[12, 14, 15, 20, 28, 84]
9=>
[10, 30, 35, 70, 90, 105]
10=>
[12, 20, 60, 70, 140, 210]
------------------------------------------------------------
(1/n)^2=(1/a1)^2+(1/a2)^2+(1/a3)^2+(1/a4)^2+(1/a5)^2+(1/a6)^2+(1/a7)^2
3=>
[4, 6, 9, 12, 36, 45, 60]
[4, 6, 10, 12, 20, 30, 60]
4=>
[5, 10, 14, 15, 28, 30, 42]
5=>
[6, 12, 20, 21, 60, 84, 105]
6=>
[9, 12, 15, 20, 36, 45, 60]
7=>
[9, 14, 28, 36, 45, 60, 84]
[10, 14, 20, 28, 30, 60, 84]
9=>
[12, 20, 21, 60, 84, 90, 105]
10=>
[12, 28, 35, 42, 70, 84, 140]
[14, 20, 30, 35, 60, 84, 140]
----------------------------------------------------------
ãŸãç«æ¹æ°ã§ã®é¢ä¿åŒã§èª¿æ»ããŠã¿ãŸããã
(1/n)^3=(1/a1)^3+(1/a2)^3+(1/a3)^3
10=>
[12, 15, 20]
*(1/10)^3=(1/12)^3+(1/15)^3+(1/20)^3 ãæç«ããããšã瀺ãã
----------------------------------------------------------------------
(1/n)^3=(1/a1)^3+(1/a2)^3+(1/a3)^3+(1/a4)^3+(1/a5)^3+(1/a6)^3+(1/a7)^3
5=>
[6, 7, 15, 21, 30, 42, 210]
6=>
[7, 10, 14, 15, 30, 42, 70]
9=>
[10, 15, 30, 36, 45, 60, 90]
10=>
[12, 14, 30, 42, 60, 84, 420]
--------------------------------------------------------------------
(1/n)^3=(1/a1)^3+(1/a2)^3+(1/a3)^3+(1/a4)^3+(1/a5)^3+(1/a6)^3+(1/a7)^3+(1/a8)^3
7=>
[9, 10, 14, 18, 63, 70, 105, 315]
9=>
[10, 14, 70, 84, 90, 105, 140, 210]
--------------------------------------------------------------------
(1/n)^3=(1/a1)^3+(1/a2)^3+(1/a3)^3+(1/a4)^3+(1/a5)^3+(1/a6)^3+(1/a7)^3+(1/a8)^3+(1/a9)^3
4=>
[5, 6, 7, 28, 35, 45, 252, 630, 1260]
5=>
[6, 7, 14, 30, 36, 42, 45, 60, 70]
6=>
[7, 10, 14, 15, 36, 42, 45, 60, 70]
9=>
[10, 15, 28, 36, 63, 70, 90, 180, 1260]
[10, 18, 20, 28, 36, 63, 70, 90, 1260]
10=>
[12, 14, 30, 42, 63, 84, 140, 180, 210]
ãªã©ãæ§æå¯èœã«ãªãããã§ãã
No.1000GAI2023幎4æ30æ¥ 10:17
(1/n)=(1/a1)+(1/a2) ã§
7â[8,56] ãšã 8â[9,72] ã¯ãªãæžãããŠããªãã®ã§ãããïŒ
äžè¬ã« nâ[n+1,n(n+1)] ã§ããã
No.1003ãããã2023幎4æ30æ¥ 17:35
7â[8,56] ãšã 8â[9,72] ããèŠéãããçç±
N=2^a*3^b*5^c*7^d
(a=0,1,2;b=0,1,2;c=0,1;d=0,1)
ãªãå åã«éå®ããïŒïŒã¿ã€ãã®æ°ã®çµã¿åãããããæ¡ä»¶ãæºããçµåãã
æ¢ãåºããŠããã®ã§ãäžèšã®æ°ã§ã®çµã¿åãããé¡ãåºããªãçµæãšãªã£ãŠããŸããã
ã§ããã8=>ã«å¯Ÿãããã¿ãŒã³ãã©ã®åéã§ãèŠéãããçµæãæããŠããŸãã
æ¢ãæ°ã®ææã
N=2^a*3^b*5^cïŒ7^d
(a=0,1,2,3;b=0,1,2;c=0,1;d=0,1)
48ãã¿ãŒã³ã§ãã£ãŠã¿ãŸããã
2=>
[3, 6]
3=>
[4, 12]
4=>
[5, 20]
[6, 12]
5=>
[6, 30]
6=>
[7, 42]
[8, 24]
[9, 18]
[10, 15]
7=>
[8, 56]
8=>
[9, 72]
[10, 40]
[12, 24]
9=>
[10, 90]
[12, 36]
10=>
[12, 60]
[14, 35]
[15, 30]
ããã§ãã£ãšå§¿ãçŸããŠããŸãã
(1/n)=(1/a1)+(1/a2)+(1/a3)+(1/a4)+(1/a5)+(1/a6)+(1/a7) ã§æ¬ æããŠããéšåã§ã
8=>
[35, 42, 60, 63, 70, 72, 84]
[40, 42, 56, 60, 63, 72, 84]
ãã®ä»å€ããçºèŠã§ããŸããã
No.1005GAI2023幎4æ30æ¥ 19:13
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No.918DD++2023幎4æ16æ¥ 08:07
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ãããç解ããŠããããã次ãèŠãŠãã ããã
--------
( a^n + b^n )^2 ⧠(a+b)^n ã®èšŒæ
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以éãaâ§2, bâ§2 ãšããã
( a^n + b^n )^2 ⧠(a+b)^n ããã
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ãã£ãŠ ( a^n + b^n )^2 ⧠(a+b)^n
-------
ãããã©ãæããŸããïŒ
No.919DD++2023幎4æ16æ¥ 08:51
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(a^n+b^n)^2-(a+b)^n=a^2n+2a^nb^n+b^2n-(a+b)^n
ãšæžãã¹ãã ã£ãã®ã§ãã(a^n+b^n)^2ïŒ(a+b)^nãå©çšããŠããç®æã¯ãããŸããã
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a^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}----(a)
ãããi=1
ãããn
b^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}----(b)
ããã i=1
ãããn
c^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}----(c)
ããã i=1
a^n+b^n=c^nãšãããšã{ãã ãa<b<cãšãã}
a^n-1+b^n-1+1=c^n-1
(a^n-1)+(b^n-1)+1=(c^n-1)
(a^n-1)+1=(c^n-1)-(b^n-1)
åŒ(a),(b),(c)ããã
n
Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}ïŒïŒ
i=1
ãn
ïŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ãi=1
ãn
ãŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}
ãi=1
ããããn
a^n-1+1ïŒÎ£ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ããããi=1
n
Σ [nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}-(a)åŒã®nCi{ã»ã»ã»}]-1----(d)
i=1
ãšãããšã(d)åŒã®
(c-1)^(n-i)-(a-1)^(n-i)
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å
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x^n-y^n=(x-y){x^(n-1)+x^(n-2)y+x^(n-3)y^2+ã»ã»ã»+xy^(n-2)+y^(n-1)}
ããã
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c-bã®é
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(c-1)^(n-i)-(a-1)^(n-i)>0
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ïŒïŒ
ãŸãã
c-1)^(n-i)-(a-1)^(n-i)
ã«ãããŠãc>b>aãããc-1>a-1ããã
c-bã®é
æ°ãšaã®é
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(c-1)^(n-i)-(a-1)^(n-i)<0
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ããããæºè¶³ããã°ããã§ã«ããŒã®æçµå®çã¯èšŒæã§ããã
ïŒïŒ
ãã ãã(d)åŒ=0ã®å ŽåãèããŠã¿ããi=1ãnã«ãããŠ
(c-1)^(n-i)-(a-1)^(n-i)
c>b>aãããc-1>a-1ããã(c-1)^(n-i)-(a-1)^(n-i)â§0----(f)
c-bã®é
æ°ãšaã®é
æ°ãåé¡ãšãªããæ¡ä»¶ã¯c-b=aãã€ãã
ïŒâïŒïŒi=nã®ãšãã
nCn{b^(n-n)+(b+1)^(n-n)+(b+2)^(n-n)+ã»ã»ã»+(c-1)^(n-n)}
-{nCn{1^(n-n)+2^(n-n)+3^(n-n)+ã»ã»ã»+(a-1)^(n-n)+1}
=c-b-a=0
ïŒâïŒïŒi=n-1ã®ãšãã
nC1{b+(b+1)+(b+2)+ã»ã»ã»+(c-1)}-{nC1{1+2+3+ã»ã»ã»+(a-1)}
=n{(c-1)c/2-(b-1)b/2-(a-1)a/2}
=n{(c-1)c-(b-1)b-(a-1)a}/2
=n{c^2-c-b^2+b-a^2+a}/2
=n{c^2-b^2-a^2-(c-b-a)}/2
c-b=aããã
=n{(c-b)(c+b)-a^2}/2
=n{a(c+b)-a^2}/2
=n{a(a+2b)-a^2}/2
=n{a^2+2ab-a^2}/2
=abn
ããã§ãa,b,nã¯èªç¶æ°ããã
nC1{b+(b+1)+(b+2)+ã»ã»ã»+(c-1)}-{nC1{1+2+3+ã»ã»ã»+(a-1)}=abn>0----(e)
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c-bã®é
æ°ãšaã®é
æ°ãåé¡ãšãªããæ¡ä»¶ã¯c-b=aãã€ãã
ããã®æå³ãããåãããªãã®ã§ããã
n
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c-bã®é
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æ°ã§aã®é
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>ïŒâïŒïŒ
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n
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ããã i=1
ãããn
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ããã i=1
ãããc-n-1ãšb^n-1ã®å·®ã¯ãc>bãªã®ã§ãïŒãb-1ãåŒãããŠã
n
Σ [nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
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ãšãªããŸãã®ã§ãïœãc-1é
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äºé
å®çããã{http://y-daisan.private.coocan.jp/html/felmer-7-2.pdfãã}
ãããn
a^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}----(a)
ããã i=1
ãããn
b^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}----(b)
ããã i=1
ãããn
c^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}----(c)
ãããi=1
a^n+b^n=c^nãšãããšã{ãã ãa<b<cãšãã}
a^n-1+b^n-1=c^n-1
(a^n-1)+(b^n-1)=(c^n-1)
(a^n-1)=(c^n-1)-(b^n-1)
åŒ(a),(b),(c)ããã
n
Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}
i=1
ãn
ïŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ãi=1
ãn
ãŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}
ãi=1
ãããn
a^n-1ïŒÎ£ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ãããi=1
ããã§ã(a^n-1)=(c^n-1)-(b^n-1)ãæãç«ã€ã«ã¯ã
ïŒïŒi=nã®ãšããa^n-1ã®nCné
ã¯ã
nCn{1^0+2^0+3^0+ã»ã»ã»+(a-1)^0)}=nCn{a-1}=a-1----(d)
äžæ¹(c^n-1)-(b^n-1)ã®nCné
ã¯ã
nCn{b^0+(b+1^0+(b+2)^0ã»ã»ã»+(c-1)^0}=nCn{(c-1)-(b-1)}=c-b---(e)
åŒ(d),(e)ãçå·ã§çµã°ããã®ã¯ã
c-b=a-1---(i)
ã®ãšãã ãã§ããã
ïŒïŒi=n-1ã®ãšããa^n-1ã®nC(n-1)é
ã¯ã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}=nC(n-1){(a-1)a/2}----(f)
äžæ¹(c^n-1)-(b^n-1)ã®nC(n-1)é
ã¯ã
nC(n-1){b+(b+1)+(b+2)ã»ã»ã»+(c-1)}=n{(c-1)c/2-(b-1)b/2}---(g)
åŒ(f),(g)ãçå·ã§çµã°ããã®ã¯ã
(c-1)c/2-(b-1)b/2=(a-1)a/2
ã®ãšãã ãã§ããã
(c-1)c-(b-1)b=(a-1)a
c^2-c-b^2+b=(a-1)a
c^2-b^2-(c-b)=(a-1)a
(c-b)(c+b-1)=(a-1)a
åŒ(d),(e)ãçãããšãåŒ(f),(g)ãçãããªããšãããªããããåŒ(i)ããã
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c+b-2=c-b
c+b-2-(c-b)=0
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2b-2=0
b=1
ããã¯ãc>b>aã«ççŸããã
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(d)â (e)ã(f)â (g)
ã€ãŸãã
(a^n-1)â (c^n-1)-(b^n-1)
a^nâ c^n-b^n
a^n+b^nâ c^n
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äºé
å®çããã{http://y-daisan.private.coocan.jp/html/felmer-7-2.pdfãã}
ãããn
a^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}----(a)
i=1
ãããn
b^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}----(b)
i=1
ãããn
c^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}----(c)
i=1
a^n+b^n=c^nãšãããšã{ãã ãa<b<cãšãã}
a^n-1+b^n-1+1=c^n-1
(a^n-1)+(b^n-1)+1=(c^n-1)
(a^n-1)+1=(c^n-1)-(b^n-1)
åŒ(a),(b),(c)ããã
n
Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}ïŒïŒ
i=1
ãn
ïŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ãi=1
ãn
ãŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}
ãi=1
ããããn
a^n-1+1ïŒÎ£ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ããããi=1
ããã§ã(a^n-1)+1=(c^n-1)-(b^n-1)ãæãç«ã€ã«ã¯ã
ïŒïŒi=nã®ãšããa^n-1ã®nCné
ã¯ã
nCn{1^0+2^0+3^0+ã»ã»ã»+(a-1)^0)}+1=nCn{a-1}+1=a----(d)
äžæ¹(c^n-1)-(b^n-1)ã®nCné
ã¯ã
nCn{b^0+(b+1^0+(b+2)^0ã»ã»ã»+(c-1)^0}=nCn{(c-1)-(b-1)}=c-b---(e)
åŒ(d),(e)ãçå·ã§çµã°ããã®ã¯ã
c-b=a---(i)
ã®ãšãã ãã§ããã
ïŒïŒi=n-1ã®ãšããa^n-1ã®nC(n-1)é
ã¯ã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}=nC(n-1){(a-1)a/2}----(f)
äžæ¹(c^n-1)-(b^n-1)ã®nC(n-1)é
ã¯ã
nC(n-1){b+(b+1)+(b+2)ã»ã»ã»+(c-1)}=n{(c-1)c/2-(b-1)b/2}---(g)
åŒ(f),(g)ãçå·ã§çµã°ããã®ã¯ã
(c-1)c/2-(b-1)b/2=(a-1)a/2
ã®ãšãã ãã§ããã
(c-1)c-(b-1)b=(a-1)a
c^2-c-b^2+b=(a-1)a
c^2-b^2-(c-b)=(a-1)a
(c-b)(c+b-1)=(a-1)a
åŒ(d),(e)ãçãããšãåŒ(f),(g)ãçãããªããšãããªããããåŒ(i)ããã
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c+b=a=c-bãåŒ(i)ãã
c+b=c-b
c+b-(c-b)=0
c+b-c+b=0
2b=0
b=0
ããã¯ãc>b>aã«ççŸããã
ãããã£ãŠã
(d)â (e)ã(f)â (g)
ã€ãŸãã
(a^n-1)â (c^n-1)-(b^n-1)
a^nâ c^n-b^n
a^n+b^nâ c^n
ãã£ãŠããã§ã«ããŒã®æçµå®çã¯åççã«èšŒæãããã
No.875ããããã¯ã¡ã¹ã2023幎4æ11æ¥ 17:48 (d) ãš (e) ãçãããšãããæ ¹æ ã¯ãªãã§ããïŒ
No.876DD++2023幎4æ11æ¥ 17:51
(a^n-1)+1=(c^n-1)-(b^n-1)ãæãç«ã€ããã§ãã
ã€ãŸãã
ããããn
a^n-1+1ïŒÎ£ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ããããi=1
ãšããããšã§ãå³èŸºã¯ãã¹ãŠæ£ã®æ°ãªã®åãªã®ã§ãã
ãŸããa^n-1+1ã¯
n
Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}ïŒïŒ
i=1
ããã¹ãŠãæ£ã®æ°ã®åã§ãããã
nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}
ã®å³å·ŠèŸºã®nCiã©ããçãããªããã°ãªããŸããã
åŒ(d),(e)ã¯ãnCnã®é
ãªã®ã§ã
a^n-1+1ã¯i=nã®
nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}ïŒïŒ
ã§ããããã®é
ã ããïŒïŒãäœåã«ããã(c^n-1)-(b^n-1)ã¯ãi=nã®
nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ã§ãããããäºãã«çãããªããã°ãªããŸãããã ããåŒ(d),(e)ã¯ãçãããªããã°ãªããŸããã
äºé
å®çã§ãåãã¹ãä¹ãªãã
(a+b)^nã®åé
ã¯ãnCi a^(n-i) b^iã§ã(a+b)^n=(c+d)^nãªãã a^(n-i) b^i= c^(n-i) d^iãšããããšã§ãã
ã€ãŸããnCiã®ä¿æ°é
ã¯a^(n-i) b^i= c^(n-i) d^iã®ããã«çãããªããªããã°ãªããªããšããããšã§ãã
No.877ããããã¯ã¡ã¹ã2023幎4æ11æ¥ 19:36
ïŒïŒïŒi=n-1ã®ãšããa^n-1ã®nC(n-1)é
ã¯ã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}=nC(n-1){(a-1)a/2}----(f)
ïŒïŒi=n-1ã®ãšããa^n-1ã®nC(n-1)é
ã¯ã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}+1=nC(n-1){(a-1)a/2}----(f)
ãããªãã§ããããã
No.878KY2023幎4æ11æ¥ 20:45
> (a+b)^n=(c+d)^nãªãã a^(n-i) b^i= c^(n-i) d^iãšããããšã§ãã
a=1, b=-1, c=0, d=0 ã§èãããšã
ã(1-1)^n = (0+0)^n ãªãã1^(n-i) (-1)^i = 0^(n-i) 0^i ãšããããšãã£ãŠæå³ã«ãªããŸããã©ããã£ãŠãŸãïŒ
No.879DD++2023幎4æ11æ¥ 21:58
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nC(n-1)=n!/(n-(n-1)!(n-1)!)=n!/(n-1)!=n
ãªã®ã§ã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}=nC(n-1){(a-1)a/2}=n{(a-1)a/2}----(f)
ãŸãã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}+1=nC(n-1){(a-1)a/2}---(f)
ïŒïŒã¯ãnCnã®é
ã ãã«äœçšããŸãã®ã§ãnC(n-1)ã«ã¯ãé¢ä¿ããŸããã
ã§ãããã
nC(n-1){1+2+3+4+5+ã»ã»ã»+(a-1)}=nC(n-1){(a-1)a/2}=n{(a-1)a/2}----(f)
ã§ããã¯ãã§ãã
DD++æ§ããã¯ããããããŸãã
ãããããµãã«ããã°ããããªããŸããã
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ïŒãªãã»ã©ãã€ãŸããã¯ã¡ã¹ããã㯠(-1)^i = 0^n ãæ£ãããšåºåŒµããŠããããã§ããïŒ
ä»åã®å Žåãa,b,c,dãšãã«ãèªç¶æ°ã§ããããããã¯ãªããªããšæããŸãã
ãææã®ã
ïŒã(1-1)^n = (0+0)^n ãªãã1^(n-i) (-1)^i = 0^(n-i) 0^i ãšããããšã
ã§ãããã(1-1)^n=0ã(0+0)^n=0ã§ãå
šäœã§èŠãã°ãçå·ãæãç«ã¡ãŸããã1^(n-i) (-1)^i = 0^(n-i) 0^iãšã¯ãèšããªãã§ããã
ã¡ãªã¿ã«ã(1-1)^nã¯ã
(1-1)^n=nC0 1^n (-1)^0+nC1 1^(n-1) (-1)^1+nC2 1^(n-2) (-1)^2+nC3 1^(n-3) (-)1^3+ã»ã»ã»ã»+nC(n-1) 1^(n-(n-1)) (-1)^(n-1)+nCn 1^(n-n) (-1)^n
ã«ãããŠã
nãå¶æ°ãªããããšãã°n=10ãªãã
0=10C0-10C1+10C2-10C3+10C4-10C5+10C6-10C7+10C8-10C9+10C10
ãã€ãã¹ã®é
ã巊蟺ã«ç§»é
ãããšã
10C1+10C3+10C5+10C7+10C9=10C0+10C2+10C4+10C6+10C8+10C10
ãã£ãŠã
nC1+nC3+nC5ã»ã»ã»+nC(n-1)=nC0+nC2+nC4+ã»ã»ã»ã»+nCn
å·Šå³ã§é
æ°ãéãã®ã«äžæè°ã«æããããããŸãããããããªã®ã§ãã
nãå¥æ°ãªããããšãã°n=11ãªãã
0=11C0-11C1+11C2-11C3+11C4-11C5+11C6-11C7+11C8-11C9+11C10-11C11
ãã€ãã¹ã®é
ã巊蟺ã«ç§»é
ãããšã
11C1+11C3+11C5+11C7+11C9+11C11=11C0+11C2+11C4+11C6+11C8+11C10
ãã£ãŠã
nC1+nC3+nC5ã»ã»ã»+nCn=nC0+nC2+nC4+ã»ã»ã»ã»+nC(n-1)
ãã¹ã«ã«ã®äžè§åœ¢ãæãåºããŠãã ããã
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ãããããããã1ã-3ã3ã-1
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ãããããã1ã-5ã10ã-10ã5ã-1
ããããã1ã-6ã15ã-20ã15ã-6ã1
ãšãªããŸãã
No.880ããããã¯ã¡ã¹ã2023幎4æ12æ¥ 07:20
ãªãã»ã©ãã€ãŸããã¯ã¡ã¹ããã㯠(-1)^i = 0^n ãæ£ãããšäž»åŒµããŠããããã§ããïŒ
No.881DD++2023幎4æ12æ¥ 07:24
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a = 1, b = 3, c = 2, d = 2 ã§èããŸãã
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ã§ã(1+3)^n = (2+2)^n ã¯æãç«ã¡ãŸãããããåé¡ãªãã§ããïŒ
ãšããããšã¯ãã¯ã¡ã¹ããã㯠1^(n-i) 3^i = 2^(n-i) 2^i ã§ããããšã
ã€ãŸã 3^i = 2^n ã¯æ£ããåŒã§ãããšäž»åŒµããããã§ããïŒ
ã¯ã¡ã¹ãããããã®åŒã誀ãã ãšæãããªãããŸã£ããåãè«çã§äœã£ã (d) = (e) ã誀ããšããããšã§ãã
No.882DD++2023幎4æ12æ¥ 16:01
DD++æ§ããã¯ããããããŸãã
ãã®ãšããã§ããã
ãã®ãã§ã«ããŒã®æçµå®çã®èšŒæã¯ãééãã§ããã
No.883ããããã¯ã¡ã¹ã2023幎4æ13æ¥ 06:59
äŒãã£ãããã§ãããã£ãã§ãã
No.884DD++2023幎4æ13æ¥ 07:48
äºé
å®çããã{http://y-daisan.private.coocan.jp/html/felmer-7-2.pdfãã}
ãããn
a^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}----(a)
i=1
ãããn
b^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}----(b)
i=1
ãããn
c^n-1=Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}----(c)
i=1
a^n+b^n=c^nãšãããšã{ãã ãa<b<cãšãã}
a^n-1+b^n-1+1=c^n-1
(a^n-1)+(b^n-1)+1=(c^n-1)
(a^n-1)+1=(c^n-1)-(b^n-1)
åŒ(a),(b),(c)ããã
n
Σ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(a-1)^(n-i)}ïŒïŒ
i=1
ãn
ïŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ãi=1
ãn
ãŒÎ£ nCi{1^(n-i)+2^(n-i)+3^(n-i)+ã»ã»ã»+(b-1)^(n-i)}
ãi=1
ããããn
a^n-1+1ïŒÎ£ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}
ããããi=1
n
Σ nCi{b^(n-i)+(b+1)^(n-i)+(b+2)^(n-i)+ã»ã»ã»+(c-1)^(n-i)}-(a)åŒ-1----(d)
i=1
ãšãããšã(d)åŒã®
(c-1)^(n-i)-(a-1)^(n-i)
ã®å€§å°é¢ä¿ã調ã¹ãã°ããã
å
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x^n-y^n=(x-y){x^(n-1)+x^(n-2)y+x^(n-3)y^2+ã»ã»ã»+xy^(n-2)+y^(n-1)}
ããã
x,yãèªç¶æ°ãªãã{}ã®äžã¯ãæ£ã®èªç¶æ°ããããã£ãŠã(x-y)ãæ£ãè² ã§x^nãšy^nã®å€§å°é¢ä¿ããããã
(c-1)^(n-i)-(a-1)^(n-i)
ã«ãããŠãc>b>aãããc-1>a-1ããã
(c-1)^(n-i)-(a-1)^(n-i)>0
ãšãªãããã£ãŠ(d)åŒã¯>0
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ããšãã°ãa=1ã®ãšãã1^3+b^3=c^3ã®ãšãb=2ã§ã3ã§ã¯ãªãã
No.898ããããã¯ã¡ã¹ã2023幎4æ15æ¥ 07:08
a,b,cã«ãããŠã
a^n+b^n=c^n
ãæãç«ã€ãšãã
(a^n+b^n)^2=(c^n)^2
ããã§ã
http://y-daisan.private.coocan.jp/html/pdf/felmer-5-4.pdfïŒç·è²ã®ããããã¯ã¡ã¹ããã¯ãªãã¯ããã°éããŸããïŒ
ã®è£é¡ããã
(a^n+b^n)^2>(a+b)^n
ã§ããããã
(a^n+b^n)^2=(c^n)^2
(a+b)^n<(c^2)^n
a,b,cã¯èªç¶æ°ããã
(a+b)<c^2
a<c^2-b
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ãšããããšã¯ãã¯ã¡ã¹ããã㯠1^(n-i) 3^i = 2^(n-i) 2^i ã§ããããšã
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