\u203b\u3053\u3053\u306b\u8a18\u8f09\u3057\u3066\u3044\u308b\u554f\u984c\u306e\u89e3\u7b54\u306f\u3042\u304f\u307e\u3067\u4e00\u4f8b\u3067\u3059\u3002
\u3000\u6570\u5b66\u3067\u306f\u69d8\u3005\u306a\u30a2\u30d7\u30ed\u30fc\u30c1\u3067\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u306e\u3067\u3001
\u3000\u3053\u306e\u89e3\u304d\u65b9\u306e\u307f\u304c\u6b63\u3068\u3044\u3046\u3053\u3068\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n\n\n\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n
\u554f\u984c1
\\(\\sqrt{n\\sqrt{n\\sqrt{n\\sqrt{n\\cdots}}}} (n>0)\\)\u3092\u7c21\u5358\u306b\u305b\u3088
<\/p>\n\n\n\n
\u89e3\u7b54
\\(\\sqrt{n\\sqrt{n\\sqrt{n\\sqrt{n\\cdots}}}}=t\\)\u3068\u7f6e\u304f\u3068\u3001
\u7121\u9650\u306b\u7d9a\u3044\u3066\u3044\u308b\u305f\u3081\u3001\\(\u4e0e\u5f0f=\\sqrt{nt}\\)\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b
\u3088\u3063\u3066\u3001\\(\u4e0e\u5f0f=t=\\sqrt{nt}\\)
\u4e21\u8fba2\u4e57\u3057\u3066
\\(t^{2}=nt\\\\
t(t-n)=0
\\require{AMSsymbols}
\\therefore t=0,n\\)
\u305f\u3060\u3057\u3001\\(n>0\\)\u3088\u308a\u3001
\\(t=0\\)\u306f\u4e0d\u9069
\\(\\require{AMSsymbols}
\\therefore \u4e0e\u5f0f=t=n\\)
<\/p>\n\n\n\n
\u554f\u984c2
\\(\\sqrt{n+\\sqrt{n+\\sqrt{n+\\sqrt{n\u2026}}}} (n>0)\\)\u3092\u7c21\u5358\u306b\u305b\u3088<\/p>\n\n\n\n
\u89e3\u7b54
\\(\\sqrt{n+\\sqrt{n+\\sqrt{n+\\sqrt{n\u2026}}}}=t\\)\u3068\u7f6e\u304f\u3068\u3001
\u7121\u9650\u306b\u7d9a\u3044\u3066\u3044\u308b\u305f\u3081\u3001\\(\u4e0e\u5f0f=\\sqrt{n+t}\\)\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b
\u3088\u3063\u3066\u3001\\(\u4e0e\u5f0f=t=\\sqrt{n+t}\\)
\u4e21\u8fba2\u4e57\u3057\u3066
\\(t^{2}=n+t\\\\
t^{2}-t-n=0\\\\
\\require{AMSmath}\\require{AMSsymbols}
t=\\dfrac{1\\pm\\sqrt{1+4n}}{2}
\\)
\u305f\u3060\u3057\u3001\\(n>0\\)\u3088\u308a\u3001
\\(\\sqrt{1+4n}>1\u304b\u3089\\\\
\\require{AMSmath}\\require{AMSsymbols}
t=\\dfrac{1-\\sqrt{1+4n}}{2}
\\)\u306f\u8ca0\u5024\u306e\u305f\u3081\u4e0d\u9069
\\(\\require{AMSsymbols}
\\therefore \u4e0e\u5f0f=t=\\dfrac{1+\\sqrt{1+4n}}{2}\\)<\/p>\n\n\n\n
\u554f\u984c3
\\(\\require{AMSmath}
a+\\dfrac{1}{a}=-1\\)\u306e\u3068\u304d\u3001
\\(3a^{5}+2a^{4}-a^{3}+a^{2}+2a+4\\)\u306e\u5024\u3092\u6c42\u3081\u3088<\/p>\n\n\n\n
\u89e3\u7b54
\\(\\require{AMSmath}
a+\\dfrac{1}{a}=-1\\)
\u4e21\u8fba\u306ba\u3092\u304b\u3051\u3066\u6574\u7406\u3059\u308b\u3068
\\(a^{2}+a+1=0\\cdots\u2460\\)
\u4e21\u8fba\u306b-1\u3092\u304b\u3051\u3066\u6574\u7406\u3059\u308b\u3068
\\(a^{3}=1\\)
\u3088\u3063\u3066\u3001
\\(3a^{5}+2a^{4}-a^{3}+a^{2}+2a+4\\\\
=3a^{2}+2a-1+a^{2}+2a+4\\\\
=4a^{2}+4a+3\\)
\u3053\u3053\u3067\u3001
\u2460\u3088\u308a
\\(a^{2}=-a-1\\)
\u3088\u3063\u3066\u3001
\\(4a^{2}+4a+3=4(-a-1)+4a+3\\\\
=-4+3=-1\\)
\\(\\therefore \u4e0e\u5f0f=-1\\)
<\/p>\n\n\n\n
\u554f\u984c4
\\(23^{2023}\\)\u3092\\(18\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u3092\u6c42\u3081\u3088<\/p>\n\n\n\n
\u89e3\u7b54
\u4ee5\u4e0b\u3059\u3079\u3066\\(\\mod18\\)\u3068\u3059\u308b
\\(23\\equiv5\\\\
23^{2}\\equiv5^2=25\\equiv7\\\\
23^{3}=23^{2}\\times23\\equiv5\\times7=35\\equiv-1\\\\
23^{6}\\equiv(23^{3})^{2}\\equiv(-1)^{2}=1\\)
\u3053\u3053\u3067\u3001
\\(2023=2022+1=6*337+1\\)
\u3088\u308a\u3001
\\(23^{2023}=(23^{6})^337\\times23\\equiv1^{337}\\times23=23\\equiv5\\)
<\/p>\n\n\n\n
\u554f\u984c5
\\(x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1=0\\)\u3092\u5b9f\u6570\u306e\u7bc4\u56f2\u3067\u56e0\u6570\u5206\u89e3\u3057\u3001\u89e3\u3092\u6c42\u3081\u3088<\/p>\n\n\n\n
\u89e3\u7b54
\u4e0e\u5f0f\u306b\\((x-1)\\)\u3092\u639b\u3051\u308b\u3068\u3001
\\(x^{12}-1=0\\\\
(x^{6})^{2}-1^{2}=0\\\\
(x^{6}-1)(x^{6}+1)=0\\\\
\\{(x^{3})^{2}-1^{2}\\}(x^{6}+1)=0\\\\
(x^{3}-1)(x^{3}+1)(x^6+1)=0\\\\
(x-1)(x^{2}+x+1)(x+1)(x^{2}-x+1)(x^6+1)=0\\\\
(x-1)(x^{2}+x+1)(x+1)(x^{2}-x+1)\\{(x^{2})^{3}+1^{3}\\}=0\\\\
(x-1)(x^{2}+x+1)(x+1)(x^{2}-x+1)(x^{2}+1)(x^{4}-x^{2}+1)=0\\\\
(x-1)(x^{2}+x+1)(x+1)(x^{2}-x+1)(x^{2}+1)\\{(x^{2})^{2}+1^{2}-x^{2}\\}=0\\\\
(x-1)(x^{2}+x+1)(x+1)(x^{2}-x+1)(x^{2}+1)\\{(x^{2}+1)^{2}-2x^{2}-x^{2}\\}=0\\\\
(x-1)(x^{2}+x+1)(x+1)(x^{2}-x+1)(x^{2}+1)\\{(x^{2}+1)^{2}-3x^{2}\\}=0\\\\
(x-1)(x^{2}+x+1)(x+1)(x^{2}-x+1)(x^{2}+1)(x^{2}+\\sqrt{3}x+1)(x^{2}-\\sqrt{3}x+1)=0\\)
\u305f\u3060\u3057\u6700\u521d\u306b(x-1)\u3092\u639b\u3051\u3066\u3044\u308b\u305f\u3081(x-1)\u306f\u9664\u5916
\\(
\\require{AMSsymbols}
\\therefore \u4e0e\u5f0f=(x+1)(x^{2}+1)(x^{2}+x+1)(x^{2}-x+1)(x^{2}+\\sqrt{3}x+1)(x^{2}-\\sqrt{3}x+1)
\\)
\u5404\u9805\u3092\u89e3\u304f\u3068
\\(\\require{AMSsymbols}
\\therefore x=-1,\\pm i,\\dfrac{-1\\pm \\sqrt{3}i}{2},\\dfrac{1\\pm \\sqrt{3}i}{2},\\dfrac{-\\sqrt{3}\\pm i}{2},\\dfrac{\\sqrt{3}\\pm i}{2}
\\)<\/p>\n\n\n\n
\u554f\u984c6
\u4efb\u610f\u306e\u6574\u6570\u306e\u5404\u6841\u306e\u548c\u304c9\u306e\u500d\u6570\u306e\u3068\u304d\u3001\u305d\u306e\u6574\u6570\u81ea\u8eab\u30829\u306e\u500d\u6570\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b<\/p>\n\n\n\n
\u89e3\u7b54
\u4efb\u610f\u306e\u6574\u6570\u3092\\(\\alpha\\)\u3068\u3059\u308b\u3068\u3001
\\(
\\alpha=10^{n}a_{n}+10^{n-1}a_{n-1}+\\cdots+10a_{1}+a_{0}
\\)
\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002
\u3053\u3053\u3067\u3001\u5404\u98051\u3064\u6e1b\u3089\u3059\u3068\u3001
\\(\\alpha=(10^{n}-1)a_{n}+(10^{n-1}-1)a_{n-1}+\\cdots+(10-1)a_{1}+a_{n}+a_{n-1}+\\cdots+a_{1}+a_{0}\\\\
=\\underbrace{999\\cdots9}_{n\u500b}a_{n}+\\underbrace{999\\cdots9}_{n-1\u500b}a_{n-1}+\\cdots+9+a_{n}+a_{n-1}+\\cdots+a_{1}+a_{0}\\\\
\\)
\u3053\u3053\u3067\u3001
\\(
\\underbrace{999\\cdots9}_{n\u500b}a_{n}+\\underbrace{999\\cdots9}_{n-1\u500b}a_{n-1}+\\cdots+9
\\)
\u306f9\u306e\u500d\u6570\u3067\u3042\u308a\u3001
\u5404\u6841\u306e\u548c\u304c9\u306e\u500d\u6570\u3088\u308a\u3001
\\(a_{n}+a_{n-1}+\\cdots+a_{1}+a_{0}\\)
\u30829\u306e\u500d\u6570\u3068\u306a\u308b\u306e\u3067\u3001
\u984c\u610f\u306f\u793a\u3055\u308c\u305f\u3002<\/p>\n\n\n\n
\u554f\u984c7
\u4efb\u610f\u306e\\(5\\)\u306e\u500d\u6570\u3067\u306a\u3044\u5947\u6570\u306f\u3001\\(1,11,111,1111,\\cdots\\)\u3068\\(1\\)\u304c\u9023\u7d9a\u3059\u308b\u6570\u306e\u3046\u3061\u3044\u305a\u308c\u304b\u306e\u7d04\u6570\u3068\u306a\u308b(\u4f55\u500d\u304b\u3059\u308c\u3070\u5fc5\u305a\\(1\\)\u304c\u9023\u7d9a\u3059\u308b\u6570\u3068\u306a\u308b)\u3053\u3068\u3092\u793a\u305b<\/p>\n\n\n\n
\u89e3\u7b54
\u4efb\u610f\u306e\\(5\\)\u306e\u500d\u6570\u3067\u306a\u3044\u5947\u6570\u3092\\(n\\)\u3068\u7f6e\u304f\\(\\cdots\\)\u2460
\\(1,11,111,1111,\\cdots\\underbrace{111\\cdots1}_{n\u500b}\\)
\u306b\u3064\u3044\u3066\u3001\u5404\u9805\u3092\\(n\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f\u6700\u5927\u3067\\(n\\)\u500b\u5b58\u5728\u3059\u308b
\u3053\u3053\u3067\u3001
\\(1,11,111,1111,\\cdots\\underbrace{111\\cdots1}_{n\u500b},\\underbrace{1111\\cdots1}_{n+1\u500b}\\)
\u306b\u3064\u3044\u3066\u540c\u69d8\u306b\\(n\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u3092\u8003\u3048\u308b\u3068\u3001
\u4f59\u308a\u304c\u540c\u3058\u7d44\u304c\u5fc5\u305a1\u3064\u5b58\u5728\u3059\u308b\u306e\u3067\u3001
\\(1,11,111,1111,\\cdots\\underbrace{111\\cdots1}_{m\u500b}\\)
\u306b\u306f\\(n\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u304c\u540c\u3058\u7d44\u304c1\u3064\u306f\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u793a\u305b\u305f
\u4f59\u308a\u304c\u540c\u3058\u7d44\u3092\u305d\u308c\u305e\u308c\\(\\alpha,\\beta(\\alpha>\\beta)\\)\u3068\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u305b\u308b
\\(\\alpha=111\\cdots1=an+r\\\\
\\beta=11\\cdots1=bn+r\\)
\\(\\alpha-\\beta\\)\u3092\u6c42\u3081\u308b\u3068\u3001
\\(111\\cdots10\\cdots0=(a+b)n\\\\
111\\cdots1\\times10\\cdots0=(a+b)n\\\\
111\\cdots1\\times2^{p}\\times5^{p}=(a+b)n\\)
\u2460\u3088\u308an\u306f2\u30685\u3068\u306f\u4e92\u3044\u306b\u7d20\u3067\u3042\u308b\u306e\u3067\u3001
\\(111\\cdots1\\)\u306fn\u3092\u7d04\u6570\u306b\u6301\u3064
\u3086\u3048\u306b\u984c\u610f\u306f\u793a\u3055\u308c\u305f\u3002<\/p>\n\n\n\n
\u554f\u984c8
\\(tan^{2}\\theta sin^{2}\\theta=tan^{2}\\theta-sin^{2}\\theta\\)
\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u305b\u3088<\/p>\n\n\n\n
\u89e3\u7b54
\\(tan\\theta=\\dfrac{sin\\theta}{cos\\theta}\u3088\u308a\\\\
tan^{2}\\theta sin^{2}\\theta=\\dfrac{sin^{2}\\theta}{cos^{2}\\theta}sin^{2}\\theta\\\\
sin^{2}\\theta=1-cos^{2}\\theta\u3088\u308a\\\\
\\dfrac{sin^{2}\\theta}{cos^{2}\\theta}sin^{2}\\theta=\\dfrac{1-cos^{2}\\theta}{cos^{2}\\theta}sin^{2}\\theta\\\\
=(\\dfrac{1}{cos^{2}\\theta}-1)sin^{2}\\theta\\\\
=\\dfrac{sin^{2}\\theta}{cos^{2}\\theta}-sin^{2}\\theta
=tan^{2}\\theta-sin^{2}\\theta\\)
\u3088\u3063\u3066\u984c\u610f\u306f\u793a\u3055\u308c\u305f\u3002<\/p>\n\n\n\n
\u554f\u984c9
\\((2023^{2024})^{2025}\u309229\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u3092\u6c42\u3081\u3088<\/p>\n\n\n\n
\u89e3\u7b54
\\(2023\u306829\\)\u306f\u4e92\u3044\u306b\u7d20\u3067\u3042\u308a\u3001\\(29\\)\u306f\u7d20\u6570
\u3088\u3063\u3066\u3001\u30d5\u30a7\u30eb\u30de\u30fc\u306e\u5c0f\u5b9a\u7406\u304b\u3089
\\(\\require{AMSmath}2023^{28}\\equiv 1 (mod \u300029) \\\\
\u4e0e\u5f0f=2023^{2024*2025}\u3088\u308a\\\\
2024*2025\\equiv 16 (mod\u300028)\\\\
\u4ee5\u964d\u306e\u5f0f\u306f(mod \u300029)\\\\
\u4e0e\u5f0f\\equiv 2023^{16}\\\\
2023\\equiv22\u3088\u308a\\\\
\u4e0e\u5f0f\\equiv22^{16}\\\\
\\equiv484^{8}\\\\
\\equiv20^{8}\\\\
\\equiv400^{4}\\\\
\\equiv23^{4}\\\\
\\equiv529^{2}\\\\
\\equiv7^{2}\\\\
\\equiv49\\\\
\\equiv20
\\)
<\/p>\n\n\n\n
\u554f\u984c10
\\(2x^{4}+5x^{3}+7x^{2}+5x+2=0\\)\u3092\u89e3\u3051<\/p>\n\n\n\n
\u89e3\u7b54
\u4e21\u8fba\u3092\\(x^{2}\\)\u3067\u5272\u308b
\\(2x^{2}+5x+7+\\dfrac{5}{x}+\\dfrac{2}{x^{2}}\\\\
=2(x^{2}+\\dfrac{1}{x^{2}})+5(x+\\dfrac{1}{x})+7\\\\
x^{2}+\\dfrac{1}{x^{2}}=(x+\\dfrac{1}{x})^{2}-2\u3088\u308a\\\\
\u4e0e\u5f0f=2(x+\\dfrac{1}{x})^{2}+5(x+\\dfrac{1}{x})+3\\\\
\u3053\u3053\u3067\u3001x+\\dfrac{1}{x}=t\u3068\u7f6e\u304f\u3068\u3001\\\\
=2t^{2}+5t+3
=(2t+3)(t+1)=0\\\\
\\require{AMSsymbols}
\\therefore t=\\dfrac{1}{2},-3\\\\
t\u3092\u5143\u306b\u623b\u3057\u3066\\\\
x+\\dfrac{1}{x}=-\\dfrac{3}{2}\\\\
x+\\dfrac{1}{x}=-1\\\\
\u5f0f\u5909\u5f62\u3057\u3066\\\\
2x^{2}+3x+2=0\\\\
x^{2}+x+1=0\\\\
\u305d\u308c\u305e\u308c\u306e\u5f0f\u3092\u89e3\u3044\u3066\\\\
\\therefore x=\\dfrac{-1\\pm\\sqrt{3}i}{2},\\dfrac{-1\\pm\\sqrt{7}i}{4}\\)<\/p>\n\n\n\n
\u554f\u984c11
\\(\\sqrt{\\dfrac{n}{\\sqrt{\\dfrac{n}{\\sqrt{\\dfrac{n}{\\sqrt{\\dfrac{n}{\\vdots}}}}}}}} (n>0)\\)\u3092\u7c21\u5358\u306b\u305b\u3088<\/p>\n\n\n\n
\u89e3\u7b54
\\(\u4e0e\u5f0f=t\\)\u3068\u7f6e\u304f\u3068\u3001
\u7121\u9650\u306b\u7d9a\u3044\u3066\u3044\u308b\u305f\u3081\u3001
\\(\\sqrt{\\dfrac{n}{t}}=t\\)
\u3068\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b
\u4e21\u8fba2\u4e57\u3057\u3066
\\(\\dfrac{n}{t}=t^{2} \\\\
n=t^{3} \\\\
t^{3}-n=0\\\\
(t-\\sqrt[3]{n})(t^{2}+\\sqrt[3]{n}t+n^{2\/3})\\\\
(t^{2}+\\sqrt[3]{n}t+n^{2\/3})\u306e\u5224\u5225\u5f0fD\u306f\\\\
D=-3n^{2\/3}<0\\\\
\u3088\u308an>0\u304b\u3089\u89e3\u306a\u3057\\\\
\u3088\u3063\u3066\\\\
t-\\sqrt[3]{n}=0\\\\
\\require{AMSsymbols}
\\therefore t=\\sqrt[3]{n}
\\)<\/p>\n\n\n\n
\u554f\u984c12
\\(Z^{2}=i\\)\u3092\u89e3\u3051<\/p>\n\n\n\n
\u89e3\u7b54
\\(Z=\\pm\\sqrt{i}\\\\
\u3053\u3053\u3067\u3001\\\\
\\sqrt{i}=a+bi\\\\
\u3068\u7f6e\u304f\\\\
(a,b\\in\\mathbb{R})\\cdots\u2460\\\\
\u4e21\u8fba\u72b62\u4e57\u3057\u3066\u3001\\\\
i=a^{2}-b^{2}+2abi\\\\
\u5b9f\u90e8\u3068\u865a\u90e8\u306e\u4fc2\u6570\u6bd4\u8f03\u3088\u308a\\\\
2ab=1\\cdots\u2461\\\\
a^{2}-b^{2}=0\\cdots\u2462\\\\
\u2461\u3088\u308a\\\\
a=\\dfrac{1}{2b}\\\\
\u2462\u306b\u4ee3\u5165\u3057\u3066\\\\
\\dfrac{1}{4b^{2}}-b^{2}=0\\\\
1-4b^{4}=0\\\\
b^{4}=\\dfrac{1}{4}\\\\
b^{2}=\\pm\\dfrac{1}{2}\\\\
\u3053\u3053\u3067\u3001\u2460\u3088\u308a\\\\
b^{2}=-\\dfrac{1}{2}\u306f\u89e3\u306b\u865a\u90e8\u304c\u542b\u307e\u308c\u308b\u306e\u3067\u4e0d\u9069\\\\
b^{2}=\\dfrac{1}{2}\\\\
b=\\pm\\dfrac{\\sqrt{2}}{2}\\\\
\u2462\u3088\u308a\\\\
a=\\pm\\dfrac{\\sqrt{2}}{2}\\\\
\\require{AMSsymbols}
\\therefore\\sqrt{i}=\\pm\\dfrac{\\sqrt{2}}{2}\\pm\\dfrac{\\sqrt{2}}{2}i\\\\
\\therefore Z=\\pm\\dfrac{\\sqrt{2}}{2}\\pm\\dfrac{\\sqrt{2}}{2}i
\\)<\/p>\n\n\n\n
\u554f\u984c13
\\(x^{3}-2x^{2}+x-1=0\u306e3\u3064\u306e\u89e3\u3092\\alpha\u3001\\beta\u3001\\gamma\u3068\u3057\u305f\u3068\u304d\u3001 \\\\
\\alpha^{5}+\\beta^{5}+\\gamma^{5}\u3068\\alpha^{-5}+\\beta^{-5}+\\gamma^{-5} \\\\
\u306e\u5024\u3092\u6c42\u3081\u3088
\\)<\/p>\n\n\n\n
\u89e3\u7b54
\\(
\\alpha\u3001\\beta\u3001\\gamma\u306f\u89e3\u3067\u3042\u308b\u306e\u3067\u3001\u4e0e\u5f0f\u306b\u4ee3\u5165\u3057\u3066\u3082\u6210\u308a\u7acb\u3064\\\\
\u4e0e\u5f0f\u306b\\alpha\u3092\u4ee3\u5165\u3057\u3001\\\\
\\alpha^{3}-2\\alpha^{2}+\\alpha-1=0\\\\
\u4e21\u8fba\u306b\\alpha^{n}\u3092\u639b\u3051\u3066\\\\
\\alpha^{n+3}-2\\alpha^{n+2}+\\alpha^{n+1}-\\alpha^{n}=0\\cdots\u2460\\\\
\\beta\u3001\\gamma\u3082\u540c\u69d8\u306b\u3057\u3066\\\\
\\beta^{n+3}-2\\beta^{n+2}+\\beta^{n+1}-\\beta^{n}=0\\cdots\u2461\\\\
\\gamma^{n+3}-2\\gamma^{n+2}+\\gamma^{n+1}-\\gamma^{n}=0\\cdots\u2462\\\\
\u2460+\u2461+\u2462\\\\
\\alpha^{n+3}+\\beta^{n+3}+\\gamma^{n+3}-2(\\alpha^{n+2}+\\beta^{n+2}+\\gamma^{n+2})+\\alpha^{n+1}+\\beta^{n+1}+\\gamma^{n+1}-(\\alpha^{n}+\\beta^{n}+\\gamma^{n})=0\\cdots\u2463\\\\
A_n=\\alpha^{n}+\\beta^{n}+\\gamma^{n}\\\\
\u3068\u7f6e\u304f\u3068\u3001\\\\
\u2463\u306f\\\\
A_{n+3}-2A_{n+2}+A_{n+1}-A_n=0\\\\
\u3068\u306a\u308a\u3001\\\\
A_{n+3}=2A_{n+2}-A_{n+1}+A_n\\cdots\u2464\\\\
A_n=A_{n+3}-2A_{n+2}+A_{n+1}\\cdots\u2465\\\\
\u3068\u306a\u308b\\\\
A_0=\\alpha^{0}+\\beta^{0}+\\gamma^{0}=1+1+1=3\\\\
\u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\u3001\\\\
\\alpha+\\beta+\\gamma=A_1=2\\\\
\\alpha\\beta+\\beta\\gamma+\\alpha\\gamma=1\\\\
\\alpha\\beta\\gamma=1\\\\
A_2=(\\alpha+\\beta+\\gamma)^{2}-2(\\alpha\\beta+\\beta\\gamma+\\alpha\\gamma)=4-2=2\\\\
\u2464\u3088\u308a\\\\
A_3=2A_2-A_1+A_0=4-2+3=5\\\\
A_4=10-2+2=10\\\\
\\therefore A_5=\\alpha^{5}+\\beta^{5}+\\gamma^{5}=20-5+2=17\\\\
\u2465\u3088\u308a\\\\
A_{-1}=A_2-2A_1+A_0=2-4+3=1\\\\
A_{-2}=2-6+1=-3\\\\
A_{-3}=3-2-3=-2\\\\
A_{-4}=1+6-2=5\\\\
\\therefore A_{-5}=\\alpha^{-5}+\\beta^{-5}+\\gamma^{-5}=-3+4+5=6
\\)<\/p>\n\n\n\n
\u554f\u984c14
\\(n\u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b\u30024^{n}-1\u304c3\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u3092\u793a\u305b\\)
\u89e3\u7b54
\u89e3\u6cd5\u304c\u591a\u6570\u5b58\u5728\u3059\u308b\u306e\u3067\u3001\u8a18\u8f09\u3067\u304d\u308b\u3060\u3051\u8a18\u8f09\u3059\u308b
\u89e3\u6cd51
\\(4\\equiv1 (mod 3)\u3088\u308a\\\\
4^{n}-1\\equiv1^{n}-1=1-1=0 (mod 3)\\\\
\u3088\u3063\u3066\u984c\u610f\u306f\u793a\u3055\u308c\u305f
\\)
\u89e3\u6cd52
\\(n=0\u306e\u6642\u30014^{0}-1=1-1=0\u3067\u984c\u610f\u3092\u6e80\u305f\u3059\\\\
n=1\u306e\u6642\u30014^{1}-1=4-1=3\u3067\u984c\u610f\u3092\u6e80\u305f\u3059\\\\
n=k\u306b\u304a\u3044\u3066\u3001\\\\
\u984c\u610f\u3092\u6e80\u305f\u3059\u3068\u4eee\u5b9a\u3057\u305f\u3068\u304d\u3001\u3059\u306a\u308f\u3061\\\\
4^{k}-1=3m (m\u306f\u4efb\u610f\u306e\u6b63\u306e\u6574\u6570)\u3068\u3057\u305f\u3068\u304d\u3001\\\\
n=k+1\u306e\u5834\u5408\\\\
4^{k}=3m+1\\\\
4^{k+1}-1=4*4^{k}-1\\\\
=4*(3m+1)-1\\\\
=12m+4-1\\\\
=12m+3\\\\
=3(4m+1)\\\\
\u3088\u3063\u3066\u984c\u610f\u306f\u793a\u3055\u308c\u305f
\\)
\u89e3\u6cd53
\\(
4^{n}-1=(2^{n}+1)(2^{n}-1)\\\\
\u3053\u3053\u3067\u30012^{n}-1\u30012^{n}\u30012^{n}+1\u306e\u9023\u7d9a3\u6570\u3092\u8003\u3048\u308b\u3068\u3001\\\\
3\u6570\u306e\u5185\u3044\u305a\u308c\u304b\u306f3\u306e\u500d\u6570\u3068\u306a\u308b\\\\
\u305f\u3060\u3057\u30012^{n}\u306f3\u306e\u500d\u6570\u3067\u306f\u306a\u3044\u306e\u3067\u3001\\\\
2^{n}-1\u30012^{n}+1\u306e\u3069\u3061\u3089\u304b\u304c3\u306e\u500d\u6570\u3068\u306a\u308b\\\\
\u3088\u3063\u3066\u984c\u610f\u306f\u793a\u3055\u308c\u305f
\\)
\u89e3\u6cd54
\\(
4^{n}-1=(2^{n}+1)(2^{n}-1)\\\\
2\\equiv-1 (mod 3)\u3088\u308a\\\\
n=\u5947\u6570\u306e\u6642\u3001\\\\
2^{n}+1\\equiv-1+1=0\\\\
n=\u5076\u6570\u306e\u6642\u3001\\\\
2^{n}-1\\equiv1-1=0\\\\
\u3088\u3063\u3066\u3001\u4efb\u610f\u306e\u6b63\u306e\u6574\u6570n\u306f3\u306e\u500d\u6570\u3067\u3042\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f
\\)
\u89e3\u6cd55
\\(
\u521d\u98051\u3001\u516c\u6bd44\u306e\u7b49\u6bd4\u6570\u5217\u306e\u548cS_{n}\u3092\u8003\u3048\u308b\\\\
S_{n}=1*(1-4^{n})\/(1-4)\\\\
=(4^{n}-1)\/(4-1)\\\\
=(4^{n}-1)\/3\\\\
\u3088\u3063\u3066\u30014^{n}-1=3S_{n}\\\\
\u3086\u3048\u306b\u984c\u610f\u306f\u793a\u3055\u308c\u305f
\\)
\u89e3\u6cd56
\\(
x^{n}-1=(x-1)(x^{n-1}+x^{n-2}+\u2026+x^{2}+x+1)\u3088\u308a\\\\
x^{n}-1\u306f(x-1)\u3092\u56e0\u6570\u306b\u6301\u3064\\\\
x=4\u306e\u6642\u3092\u8003\u3048\u308b\u3068\u3001\\\\
4^{n}-1\u306f4-1=3\u3092\u56e0\u6570\u306b\u6301\u3064\\\\
\u3088\u3063\u3066\u984c\u610f\u306f\u793a\u3055\u308c\u305f
\\)
\u89e3\u6cd57
\\(
4=3+1\\\\
4^{n}=(3+1)^{n}\\\\
\u4e8c\u9805\u5c55\u958b\u3059\u308b\u3068\\\\
=3(\uff5e)+1
\u3088\u3063\u3066\u3001
4^{n}-1=3(\uff5e)
\u3086\u3048\u306b\u984c\u610f\u306f\u793a\u3055\u308c\u305f
\\)<\/p>\n","protected":false},"excerpt":{"rendered":"
\u203b\u3053\u3053\u306b\u8a18\u8f09\u3057\u3066\u3044\u308b\u554f\u984c\u306e\u89e3\u7b54\u306f\u3042\u304f\u307e\u3067\u4e00\u4f8b\u3067\u3059\u3002\u3000\u6570\u5b66\u3067\u306f\u69d8\u3005\u306a\u30a2\u30d7\u30ed\u30fc\u30c1\u3067\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u306e\u3067\u3001\u3000\u3053\u306e\u89e3\u304d\u65b9\u306e\u307f\u304c\u6b63\u3068\u3044\u3046\u3053\u3068\u306f\u3042\u308a\u307e\u305b\u3093\u3002 \u554f\u984c1\u3092\u7c21\u5358\u306b\u305b\u3088 \u89e3\u7b54\u3068\u7f6e\u304f\u3068\u3001\u7121\u9650\u306b\u7d9a\u3044\u3066\u3044\u308b\u305f\u3081\u3001\u3068\u3059\u308b\u3053\u3068 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":156,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-blank","meta":{"footnotes":""},"class_list":["post-5","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.kokosan60.com\/sonota\/suugaku\/mondai\/index.php?rest_route=\/wp\/v2\/pages\/5","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.kokosan60.com\/sonota\/suugaku\/mondai\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.kokosan60.com\/sonota\/suugaku\/mondai\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.kokosan60.com\/sonota\/suugaku\/mondai\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kokosan60.com\/sonota\/suugaku\/mondai\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5"}],"version-history":[{"count":181,"href":"https:\/\/www.kokosan60.com\/sonota\/suugaku\/mondai\/index.php?rest_route=\/wp\/v2\/pages\/5\/revisions"}],"predecessor-version":[{"id":263,"href":"https:\/\/www.kokosan60.com\/sonota\/suugaku\/mondai\/index.php?rest_route=\/wp\/v2\/pages\/5\/revisions\/263"}],"up":[{"embeddable":true,"href":"https:\/\/www.kokosan60.com\/sonota\/suugaku\/mondai\/index.php?rest_route=\/wp\/v2\/pages\/156"}],"wp:attachment":[{"href":"https:\/\/www.kokosan60.com\/sonota\/suugaku\/mondai\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}