{"id":169,"date":"2025-07-03T12:52:02","date_gmt":"2025-07-03T03:52:02","guid":{"rendered":"https:\/\/math-friend.com\/?p=169"},"modified":"2025-08-01T09:20:39","modified_gmt":"2025-08-01T00:20:39","slug":"%e3%80%90%e9%a6%99%e5%b7%9d%e5%a4%a7%e5%ad%a6%e3%80%91%e5%b8%b0%e7%b4%8d%e6%b3%95%e3%82%92%e4%bd%bf%e3%81%a3%e3%81%a6%e4%b8%8d%e7%ad%89%e5%bc%8f%e3%81%ae%e8%a8%bc%e6%98%8e","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=169","title":{"rendered":"\u3010\u9999\u5ddd\u5927\u5b66\u5165\u8a66\u3011\u5e30\u7d0d\u6cd5\u3092\u4f7f\u3063\u3066\u4e0d\u7b49\u5f0f\u306e\u8a3c\u660e(2025)"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\\(x\\)\u304c\\(0<x<1\\)\u3092\u307f\u305f\u3059\u3068\u304d, \u4ee5\u4e0b\u306e\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.
(1) \\(1+x+x^2<2+x^3\\)
(2) \\(1+x+x^2+\\cdots+x^n<n+x^{n+1}\\) (\u305f\u3060\u3057, \\(n\\)\u306f2\u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b)
\u30d2\u30f3\u30c8: \\(n\\)\u306b\u95a2\u3059\u308b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u308b.
(2025 \u9999\u5ddd\u5927\u5b66)<\/span><\/p>\n\n\n\n

\u4eca\u56de\u306e\u554f\u984c\u306f\u7d50\u5c40\u306f\\(n\\)\u304c2\u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3067, \\(0<x<1\\)\u306e\u3068\u304d, \u4e0d\u7b49\u5f0f\\(1+x+x^2+\\cdots+x^n<n+x^{n+1}\\)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3059\u554f\u984c\u3067\u3059. \u89e3\u6cd5\u306e\u30d2\u30f3\u30c8\u3068\u3057\u3066\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u4f7f\u3046\u3053\u3068\u304c\u660e\u793a\u3055\u308c, \u307e\u305f\u51fa\u767a\u70b9\u3068\u306a\u308b\\(n=2\\)\u306e\u3068\u304d\u306e\u8a3c\u660e\u304c(1)\u3067\u8a98\u5c0e\u3055\u308c\u3066\u3044\u308b\u3068\u3044\u3046\u3068\u3066\u3082\u89aa\u5207\u306a\u554f\u984c\u3067\u3059.

\u3053\u306e\u554f\u984c\u306e\u3088\u3046\u306b\\(x\\)\u304c\u5165\u3063\u305f\u4e0d\u7b49\u5f0f\u3092\u8a3c\u660e\u3059\u308b\u969b\u306f, \u9805\u3092\u5de6\u8fba\u304b\u53f3\u8fba\u306b\u5bc4\u305b\u3066\\(x\\)\u306e\u95a2\u6570\u3068\u307f\u306a\u3057, \\(x\\)\u306e\u7bc4\u56f2\u5185\u3067\u305d\u306e\u95a2\u6570\u304c\\(0\\)\u4ee5\u4e0a\u3001\\(0\\)\u3088\u308a\u5927\u304d\u3044, \\(0\\)\u4ee5\u4e0b, \\(0\\)\u672a\u6e80\u306a\u3069\u3092\u793a\u3057\u3066\u3044\u3051\u3070\u826f\u3044\u3067\u3059.

\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\n

(1) \u8a3c\u660e\u3057\u305f\u3044\u4e0d\u7b49\u5f0f
$$1+x+x^2<2+x^3$$
\u3092\u5909\u5f62\u3059\u308b\u3068,
$$2+x^3-(1+x+x^2)>0$$
\u3068\u306a\u308b. \u5143\u306e\u4e0d\u7b49\u5f0f\u306e\u8a3c\u660e\u306e\u305f\u3081\u306b\u306f, \u3053\u306e\u5de6\u8fba\u3092\\(f(x)=2+x^3-(1+x+x^2)=x^3-x^2-x+1\\)\u3068\u304a\u304d, \\(0<x<1\\)\u306e\u3068\u304d, \\(f(x)>0\\)\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3059\u308c\u3070\u826f\u3044. \\(f(x)\\)\u306e\u5897\u6e1b\u3092\u8abf\u3079\u308b\u305f\u3081\u306b, \u5fae\u5206\u3059\u308b\u3068,
$$f^\\prime(x)=3x^2-2x-1=(3x+1)(x-1)$$
\u3068\u306a\u308a, \\(f^\\prime(x)=0\\)\u3068\u304a\u304f\u3068,
$$x=-\\frac{1}{3}, 1$$\u3068\u306a\u308b. \\(0<x<1\\)\u306e\u7bc4\u56f2\u3067\u5897\u6e1b\u8868\u3092\u66f8\u304f\u3068,
$$
\\begin{array}{|c|c|c|c|}
\\hline
x & 0 & \\cdots & 1 \\\\
\\hline
f'(x) & – & – & 0 \\\\
\\hline
f(x) & 1 & \\searrow & 0 \\\\
\\hline
\\end{array}
$$
\u3068\u306a\u308a, \\(f(x)\\)\u306f\\(0<x<1\\)\u3067\u5358\u8abf\u6e1b\u5c11\u3067\u3042\u308a, \\(f(1)=0\\)\u304b\u3089\\(0<x<1\\)\u306e\u7bc4\u56f2\u3067\\(f(x)>0\\)\u3067\u3042\u308b\u3053\u3068\u304c\u8a00\u3048\u308b.<\/p>\n\n\n\n

(2) \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u8a3c\u660e\u3059\u308b. \u51fa\u767a\u70b9\u3068\u306a\u308b\\(n=2\\)\u306f(1)\u3067\u793a\u3057\u3066\u3044\u308b.

\\(n=k\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3059\u308b. \u3064\u307e\u308a\\(0<x<1\\)\u306e\u7bc4\u56f2\u3067\u4ee5\u4e0b\u306e\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3068\u3059\u308b.
$$
1+x+x^2+\\cdots+x^k<k+x^{k+1} \\,\\,\\,\\,\\text{\u30fb\u30fb\u30fb\u2460}
$$
\u3053\u306e\u524d\u63d0\u306e\u4e0b, \\(n=k+1\\)\u306e\u3068\u304d\u3082\u4e0d\u7b49\u5f0f,
$$
1+x+x^2+\\cdots+x^k + x^{k+1}<(k+1)+x^{(k+1)+1}
$$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b.

\u8a3c\u660e\u3057\u305f\u3044\u4e0d\u7b49\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068,
$$
(k+1)+x^{k+2}-( 1+x+x^2+\\cdots+x^k + x^{k+1})>0
$$
\u3068\u306a\u308b\u306e\u3067,
$$g(x)=(k+1)+x^{k+2}-( 1+x+x^2+\\cdots+x^k + x^{k+1})
$$\u3068\u304a\u304d, \\(0<x<1\\) \u306e\u3068\u304d, \\(g(x)>0\\)\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b.

\u4ee5\u4e0b\u306e\u3088\u3046\u306b\\(g(x)\\)\u3092\u5909\u5f62\u3057, \u2460\u306e\u4e0d\u7b49\u5f0f\u3092\u7528\u3044\u308b\u3068,
$$
\\begin{align}
g(x)&=(k+1)+x^{k+2}-x^{k+1}-( 1+x+x^2+\\cdots+x^k)\\\\[1.5ex]
&>(k+1)+x^{k+2}-x^{k+1}-(k+x^{k+1})\\\\[1.5ex]
&=x^{k+2}-2x^{k+1}+1
\\end{align}$$
\u3068\u306a\u308b\u306e\u3067,
$$h(x)=x^{k+2}-2x^{k+1}+1$$\u3068\u304a\u304d, \\(0<x<1\\)\u306e\u7bc4\u56f2\u3067\\(h(x)>0\\)\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3059\u308c\u3070, \u305d\u308c\u306f\u540c\u7bc4\u56f2\u3067\\(g(x)>0\\)\u3092\u8a3c\u660e\u3057\u305f\u3053\u3068\u3068\u306a\u308b.

\\(0<x<1\\)\u3067\u306e\\(h(x)\\)\u306e\u5897\u6e1b\u3092\u8abf\u3079\u308b\u305f\u3081\u5fae\u5206\u3059\u308b\u3068,
$$
h^\\prime(x)=(k+2)x^{k+1}-2(k+1)x^k=(k+2)x^k\\left(x-\\frac{2(k+1)}{k+2}\\right)
$$
\u3068\u306a\u308a, \\(h^\\prime(x)=0\\)\u3068\u304a\u304f\u3068, \\(x=0\\)\u307e\u305f\u306f, \\(x=\\frac{2(k+1)}{k+2}\\)\u3067\u3042\u308b. \u3053\u3053\u3067,
$$
\\frac{2(k+1)}{k+2}=\\frac{(k+2)+k}{k+2}=1+\\frac{k}{k+2}>1
$$
\u3067\u3042\u308b\u304b\u3089, \\(0<x<1\\)\u306e\u7bc4\u56f2\u3067\\(h(x)\\)\u306e\u5897\u6e1b\u8868\u3092\u66f8\u304f\u3068,
$$
\\begin{array}{|c|c|c|c|}
\\hline
x & 0 & \\cdots & 1 \\\\
\\hline
h'(x) & 0 & – & – \\\\
\\hline
h(x) & 1 & \\searrow & 0 \\\\
\\hline
\\end{array}
$$
\u3068\u306a\u308b. \u3088\u3063\u3066, \\(h(x)\\)\u306f\\(0<x<1\\)\u3067\u5358\u8abf\u6e1b\u5c11\u3067\u3042\u308a, \\(h(1)=0\\)\u304b\u3089\\(0<x<1\\)\u306e\u7bc4\u56f2\u3067\\(h(x)>0\\)\u3067\u3042\u308b\u3053\u3068\u304c\u8a00\u3048\u308b. \u3057\u305f\u304c\u3063\u3066, \\(0<x<1\\)\u306e\u7bc4\u56f2\u3067\\(g(x)>0\\)\u3067\u3042\u308b\u3053\u3068\u3082\u308f\u304b\u308a, \\(n=k+1\\)\u3067\u3082\u4e0e\u3048\u3089\u308c\u305f\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u8a3c\u660e\u3055\u308c\u305f.

\u4ee5\u4e0a\u304b\u3089\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u308a, \\(n\\)\u304c\\(2\\)\u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u306e\u3068\u304d\u306b,
$$
1+x+x^2+\\cdots+x^n<n+x^{n+1}
$$
\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

(1)\u306f\u57fa\u672c\u554f\u984c\u3067\u3059\u306d. (2)\u306f\\(g(x)\\)\u306e\u5897\u6e1b\u3092\u8abf\u3079\u308b\u305f\u3081\u306b, \\(g(x)\\)\u3092\u5fae\u5206\u3059\u308b\u3068\\(k+1\\)\u6b21\u65b9\u7a0b\u5f0f\u304c\u51fa\u3066\u304f\u308b\u306e\u3067\u3064\u307e\u308a\u307e\u3059. \u5fae\u5206\u524d\u306b\u5e30\u7d0d\u6cd5\u306e\u524d\u63d0\u3068\u306a\u308b\u4e0d\u7b49\u5f0f\u3067\u7c21\u5358\u306a\u5f62\u306e\u95a2\u6570\u306b\u3057\u3066\u304a\u304f\u3053\u3068\u304c\u30df\u30bd\u3067\u3059.

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n