{"id":12224,"date":"2023-01-14T16:13:06","date_gmt":"2023-01-14T07:13:06","guid":{"rendered":"https:\/\/hatsudy.com\/jp\/?p=12224"},"modified":"2023-01-21T00:39:54","modified_gmt":"2023-01-20T15:39:54","slug":"mathematical-induction","status":"publish","type":"post","link":"https:\/\/hatsudy.com\/jp\/mathematical-induction.html","title":{"rendered":"\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\uff1a\u6570\u5217\u306e\u516c\u5f0f\u3084\u6f38\u5316\u5f0f\u3001\u4e0d\u7b49\u5f0f\u306e\u8a3c\u660e"},"content":{"rendered":"\n<p>\u6570\u5217\u304c\u95a2\u308f\u308b\u8a3c\u660e\u6cd5\u306b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u304c\u3042\u308a\u307e\u3059\u3002\\(n=1\\)\u3068\\(n=k+1\\)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u3059\u3079\u3066\u306e\u6761\u4ef6\u304c\u6210\u7acb\u3059\u308b\u3068\u8a3c\u660e\u3067\u304d\u308b\u65b9\u6cd5\u304c\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u3059\u3002<\/p>\n<p>\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u5229\u7528\u3057\u3066\u8a3c\u660e\u3059\u308b\u3068\u304d\u3001\u5b9f\u969b\u306b\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u3067\u6163\u308c\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002\u6f38\u5316\u5f0f\u3068\u540c\u69d8\u306b\u554f\u984c\u306e\u89e3\u304d\u65b9\u306f\u6c7a\u307e\u3063\u3066\u304a\u308a\u3001\u3069\u306e\u3088\u3046\u306a\u4ed5\u7d44\u307f\u306b\u3088\u3063\u3066\u8a3c\u660e\u3067\u304d\u308b\u306e\u304b\u5b66\u3076\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n<p>\u306a\u304a\u6f38\u5316\u5f0f\u3067\u306f\u5fdc\u7528\u554f\u984c\u3092\u89e3\u304b\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u30b1\u30fc\u30b9\u3082\u3042\u308a\u307e\u3059\u3002\u6570\u5217\u3084\u6f38\u5316\u5f0f\u306b\u306f\u96e3\u554f\u304c\u305f\u304f\u3055\u3093\u5b58\u5728\u3057\u307e\u3059\u3002\u3053\u308c\u306f\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3082\u540c\u69d8\u3067\u3042\u308a\u3001\u5fdc\u7528\u554f\u984c\u306f\u89e3\u304f\u306e\u304c\u96e3\u3057\u3044\u3067\u3059\u3002\u305d\u3053\u3067\u57fa\u672c\u3092\u5229\u7528\u3057\u3001\u5fdc\u7528\u554f\u984c\u3092\u89e3\u3051\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u3069\u306e\u3088\u3046\u306b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u5229\u7528\u3057\u3066\u8a3c\u660e\u3059\u308c\u3070\u3044\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u8907\u6570\u306e\u4f8b\u984c\u3092\u5229\u7528\u3057\u3066\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306e\u5229\u7528\u6cd5\u3092\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<h2>\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u308b\u8a3c\u660e\u306e\u65b9\u6cd5<\/h2>\n<p>\u30c9\u30df\u30ce\u3092\u30a4\u30e1\u30fc\u30b8\u3059\u308c\u3070\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7406\u89e3\u3067\u304d\u307e\u3059\u3002\u30c9\u30df\u30ce\u304c\u5012\u308c\u308b\u305f\u3081\u306b\u306f\u3001\u6700\u521d\u306e\u304d\u3063\u304b\u3051\u304c\u5fc5\u8981\u3067\u3059\u3002\u6700\u521d\u306e\u30c9\u30df\u30ce\u3092\u5012\u305b\u3070\u3001\u30c9\u30df\u30ce\u306f\u6700\u5f8c\u307e\u3067\u5012\u308c\u307e\u3059\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/hatsudy.com\/jp\/wp-content\/uploads\/2023\/01\/djdhq1.jpg\" alt=\"\" width=\"600\" height=\"214\" class=\"aligncenter size-full wp-image-12227\" \/><\/p>\n<p>\u30c9\u30df\u30ce\u3068\u540c\u3058\u6982\u5ff5\u304c\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u3059\u3002\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u306f\u3001\u307e\u305a\\(n=1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002\u305d\u306e\u5f8c\u3001\\(n=k\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3001\\(n=k+1\\)\u3067\u3042\u3063\u3066\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3057\u307e\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(n=1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064<\/li>\n<li>\\(n=1\\)\u3067\u6210\u308a\u7acb\u3064\u305f\u3081\u3001\\(n=2\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064<\/li>\n<li>\\(n=2\\)\u3067\u6210\u308a\u7acb\u3064\u305f\u3081\u3001\\(n=3\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064<\/li>\n<li>\\(n=3\\)\u3067\u6210\u308a\u7acb\u3064\u305f\u3081\u3001\\(n=4\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064<\/li>\n<li>\u2026\u2026<\/li>\n<\/ul>\n<p><span style=\"color: #0000ff;\">\\(n=k+1\\)\u3067\u3042\u3063\u3066\u3082\u6210\u308a\u7acb\u3064\u305f\u3081\u3001\u30c9\u30df\u30ce\u5012\u3057\u306e\u3088\u3046\u306b\u3059\u3079\u3066\u306e\u6761\u4ef6\u3067\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3067\u304d\u307e\u3059<\/span>\u3002\u305f\u3060\u6700\u521d\u306e\u304d\u3063\u304b\u3051\u304c\u5fc5\u9808\u3067\u3042\u308a\u3001\\(n=1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u8a3c\u660e\u3067\u304d\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u3001\\(n=1\\)\u3068\\(n=k+1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b\u306e\u3067\u3059\u3002<\/p>\n<h3>\u57fa\u672c\u7684\u306a\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306e\u8a08\u7b97<\/h3>\n<p>\u305d\u308c\u3067\u306f\u3001\u5b9f\u969b\u306b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u8a3c\u660e\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002\\(n\\)\u304c\u81ea\u7136\u6570\u306e\u3068\u304d\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u4ee5\u4e0b\u306e\u7b49\u5f0f\u3092\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(1^2+2^2+3^2+\\)\\(&#8230;+n^2\\)\\(=\\displaystyle\\frac{1}{6}n(n+1)(2n+1)\\)<\/li>\n<\/ul>\n<p>\u6570\u5217\u3067\u306f\u3001\u4e8c\u4e57\u306e\u548c\u306e\u516c\u5f0f\u3092\u899a\u3048\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002\u305f\u3060\u3001\u306a\u305c\u3053\u306e\u516c\u5f0f\u304c\u6210\u308a\u7acb\u3064\u306e\u304b\u7406\u89e3\u3057\u3066\u3044\u306a\u3044\u4eba\u306f\u591a\u3044\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p><strong>\u30fb[1] \\(n=1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u8a3c\u660e\u3059\u308b<\/strong><\/p>\n<p>\\(n=1\\)\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(\\displaystyle\\frac{1}{6}\u00d71\u00d72\u00d73=1^2\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\\(n=1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u8a3c\u660e\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<p><strong>\u30fb[2] \\(n=k\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3001\\(n=k+1\\)\u3067\u3042\u3063\u3066\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b<\/strong><\/p>\n<p>\\(n=k\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3059\u308b\u3068\u3001\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(1^2+2^2+3^2+\\)\\(&#8230;+k^2\\)\\(=\\displaystyle\\frac{1}{6}k(k+1)(2k+1)\\)<\/li>\n<\/ul>\n<p>\u305d\u3053\u3067\u3001\\(n=k+1\\)\u306e\u3068\u304d\u3092\u8003\u3048\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(1^2+2^2+3^2+\\)\\(&#8230;+k^2\\)\\(+(k+1)^2\\)<\/p>\n<p>\\(=\\displaystyle\\frac{1}{6}k(k+1)(2k+1)\\)\\(+(k+1)^2\\)<\/p>\n<p>\\(=\\displaystyle\\frac{1}{6}(k+1)\\{k(2k+1)+6(k+1)\\}\\)<\/p>\n<p>\\(=\\displaystyle\\frac{1}{6}(k+1)(2k^2+7k+6)\\)<\/p>\n<p>\\(=\\displaystyle\\frac{1}{6}(k+1)(k+2)(2k+3)\\)<\/p>\n<p>\u5de6\u8fba\u306e\\(n\\)\u3092\\(k+1\\)\u306b\u5909\u3048\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u53f3\u8fba\u306e\\(n\\)\u3092\\(k+1\\)\u306b\u5909\u3048\u305f\u3068\u304d\u306e\u5f0f\u3092\u5c0e\u304d\u51fa\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u3064\u307e\u308a\u3001\\(n=k+1\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3061\u307e\u3059\u3002[1]\u3068[2]\u3088\u308a\u3001\u3059\u3079\u3066\u306e\u81ea\u7136\u6570\\(n\\)\u3067\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n<h3>\u6f38\u5316\u5f0f\u3067\u4e00\u822c\u9805\u3092\u63a8\u6e2c\u3057\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u3063\u3066\u8a3c\u660e\u3059\u308b<\/h3>\n<p>\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u6f38\u5316\u5f0f\u3092\u7528\u3044\u308b\u30b1\u30fc\u30b9\u3082\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u306f\u6570\u5217\u306e\u4e00\u822c\u9805\u3092\u63a8\u6e2c\u3057\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u3063\u3066\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002\u4ee5\u4e0b\u306e\u554f\u984c\u306e\u7b54\u3048\u306f\u4f55\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<ul>\n<li>\\(a_1=2\\)\u3067\u3042\u308b\u6570\u5217\\(\\{a_n\\}\\)\u306b\u3064\u3044\u3066\u3001\u3059\u3079\u3066\u306e\u81ea\u7136\u6570\\(n\\)\u306b\u5bfe\u3057\u3066\\(a_{n+1}=2-\\displaystyle\\frac{1}{a_n}\\)\u3092\u6e80\u305f\u3057\u3066\u3044\u307e\u3059\u3002\\(a_2\\)\u3001\\(a_3\\)\u3001\\(a_4\\)\u3092\u8a08\u7b97\u3057\u3001\u4e00\u822c\u9805\u3092\u63a8\u6e2c\u3057\u305f\u5f8c\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u3063\u3066\u305d\u306e\u63a8\u6e2c\u304c\u6b63\u3057\u3044\u3053\u3068\u3092\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\u307e\u305a\\(a_2\\)\u3001\\(a_3\\)\u3001\\(a_4\\)\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(a_2=2-\\displaystyle\\frac{1}{2}=\\displaystyle\\frac{3}{2}\\)<\/li>\n<li>\\(a_3=2-\\displaystyle\\frac{2}{3}=\\displaystyle\\frac{4}{3}\\)<\/li>\n<li>\\(a_4=2-\\displaystyle\\frac{3}{4}=\\displaystyle\\frac{5}{4}\\)<\/li>\n<\/ul>\n<p>\\(a_1=\\displaystyle\\frac{2}{1}\\)\u3068\u8003\u3048\u308b\u3068\u3001\u5206\u6bcd\u306f\\(n\\)\u3001\u5206\u5b50\u306f\\(n+1\\)\u3067\u3042\u308a\u3001\\(a_n=\\displaystyle\\frac{n+1}{n}\\)\u3068\u63a8\u6e2c\u3067\u304d\u307e\u3059\u3002\u305d\u308c\u3067\u306f\u3001\u3053\u306e\u63a8\u6e2c\u304c\u6b63\u3057\u3044\u3053\u3068\u3092\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p><strong>\u30fb[1] \\(n=1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u8a3c\u660e\u3059\u308b<\/strong><\/p>\n<p>\\(n=1\\)\u306e\u3068\u304d\u3001\\(a_1=\\displaystyle\\frac{1+1}{1}=2\\)\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\\(n=1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n<p><strong>\u30fb[2] \\(n=k\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3001\\(n=k+1\\)\u3067\u3042\u3063\u3066\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b<\/strong><\/p>\n<p>\\(n=k\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3059\u308b\u3068\u3001\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_k=\\displaystyle\\frac{k+1}{k}\\)<\/li>\n<\/ul>\n<p>\u305d\u3053\u3067\u3001\\(a_k=\\displaystyle\\frac{k+1}{k}\\)\u3092\\(a_{k+1}=2-\\displaystyle\\frac{1}{a_k}\\)\u306b\u4ee3\u5165\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(a_{k+1}=2-\\displaystyle\\frac{1}{a_k}\\)<\/p>\n<p>\\(a_{k+1}=2-\\displaystyle\\frac{k}{k+1}\\)<\/p>\n<p>\\(a_{k+1}=\\displaystyle\\frac{k+2}{k+1}\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\\(n=k+1\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3061\u307e\u3059\u3002[1]\u3068[2]\u3088\u308a\u3001\u4e00\u822c\u9805\u306f\\(a_n=\\displaystyle\\frac{n+1}{n}\\)\u3067\u3059\u3002<\/p>\n<h2>\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u305f\u4e0d\u7b49\u5f0f\u306e\u8a3c\u660e<\/h2>\n<p>\u6b21\u306b\u3001\u4e0d\u7b49\u5f0f\u3092\u542b\u3080\u5834\u5408\u306e\u8a3c\u660e\u65b9\u6cd5\u3092\u5b66\u3073\u307e\u3057\u3087\u3046\u3002\u4e0d\u7b49\u5f0f\u3092\u8a3c\u660e\u3059\u308b\u65b9\u6cd5\u3068\u3057\u3066\u306f\u3001\u5f15\u304d\u7b97\u3092\u5229\u7528\u3057\u307e\u3059\u3002\u4f8b\u3048\u3070\\(a&gt;b\\)\u3092\u8a3c\u660e\u3057\u305f\u3044\u5834\u5408\u3001\\(a-b&gt;0\\)\u3067\u3042\u308b\u3068\u793a\u305b\u3070\u3044\u3044\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u4ee5\u4e0b\u306e\u554f\u984c\u3092\u89e3\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\u81ea\u7136\u6570\\(n\\)\u304c\\(n\u22675\\)\u3092\u6e80\u305f\u3059\u3068\u304d\u3001\\(n^2&lt;2^n\\)\u3092\u6e80\u305f\u3059\u3053\u3068\u3092\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\\(n^2&lt;2^n\\)\u3092\u8a3c\u660e\u3059\u308b\u305f\u3081\u306b\u306f\u3001\\(0&lt;2^n-n^2\\)\u3092\u8a3c\u660e\u3059\u308c\u3070\u3044\u3044\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p><strong>\u30fb[1] \\(n=5\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u8a3c\u660e\u3059\u308b<\/strong><\/p>\n<p>\\(0&lt;2^5-5^2\\)<\/p>\n<p>\\(0&lt;32-25\\)<\/p>\n<p>\\(0&lt;7\\)<\/p>\n<p>\u305d\u306e\u305f\u3081\u3001\\(n=5\\)\u306e\u3068\u304d\u306b\u4e0d\u7b49\u5f0f\u306f\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n<p><strong>\u30fb[2] \\(n=k\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3001\\(n=k+1\\)\u3067\u3042\u3063\u3066\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b<\/strong><\/p>\n<p>\\(n=k\\)\uff08\\(k\u22675\\)\uff09\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3059\u308b\u3068\u3001\u5f0f\u306f\\(k^2&lt;2^k\\)\u3068\u306a\u308a\u307e\u3059\u3002\u305d\u3053\u3067\\(n=k+1\\)\u306e\u3068\u304d\u3092\u8003\u3048\u3066\\((k+1)^2&lt;2^{k+1}\\)\u3001\u3064\u307e\u308a\\(0&lt;2^{k+1}-(k+1)^2\\)\u3092\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(2^{k+1}-(k+1)^2\\)<\/p>\n<p>\\(=2\u00b72^k-(k+1)^2\\)<\/p>\n<p>\u306a\u304a\\(k^2&lt;2^k\\)\u304c\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3066\u3044\u308b\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u4e0d\u7b49\u5f0f\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<p>\\(2\u00b7k^2-(k+1)^2&lt;2\u00b72^k-(k+1)^2\\)<\/p>\n<p>\u305d\u306e\u305f\u3081\\(0&lt;2\u00b7k^2-(k+1)^2\\)\u3092\u8a3c\u660e\u3067\u304d\u308c\u3070\u3001\\(0&lt;2\u00b7k^2-(k+1)^2\\)\\(&lt;2^{k+1}-(k+1)^2\\)\u3068\u306a\u308b\u305f\u3081\u3001\\(0&lt;2^{k+1}-(k+1)^2\\)\u3092\u8a3c\u660e\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067\u3001\\(2\u00b7k^2-(k+1)^2\\)\u306b\u7740\u76ee\u3057\u3066\u5f0f\u3092\u5909\u5f62\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(2\u00b7k^2-(k+1)^2\\)<\/p>\n<p>\\(=k^2-2k-1\\)<\/p>\n<p>\\(=(k-1)^2-2\\)<\/p>\n<p>\\(k\u22675\\)\u3067\u3042\u308b\u305f\u3081\u3001\\(y=(k-1)^2-2\\)\u3067\\(y\\)\u306f\u5fc5\u305a\u6b63\u306e\u5024\u306b\u306a\u308a\u307e\u3059\u3002\u305d\u306e\u305f\u3081\\(k\u22675\\)\u306e\u3068\u304d\u3001\\(0&lt;2\u00b7k^2-(k+1)^2\\)\u3067\u3059\u3002\u3053\u3046\u3057\u3066\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n<p>\\(0&lt;2\u00b7k^2-(k+1)^2\\)\\(&lt;2^{k+1}-(k+1)^2\\)<\/p>\n<p>\\(0&lt;2^{k+1}-(k+1)^2\\)<\/p>\n<p>\\((k+1)^2&lt;2^{k+1}\\)<\/p>\n<p>\u3064\u307e\u308a\u3001\\(n=k+1\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3061\u307e\u3059\u3002[1]\u3068[2]\u3088\u308a\u3001\\(n\u22675\\)\u3092\u6e80\u305f\u3059\u3059\u3079\u3066\u306e\u81ea\u7136\u6570\\(n\\)\u3067\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u4eca\u56de\u306e\u8a3c\u660e\u554f\u984c\u3067\u306f\\(n=1\\)\u3067\u306f\u306a\u304f\u3001\\(n=5\\)\u304c\u30b9\u30bf\u30fc\u30c8\u5730\u70b9\u3067\u3059\u3002\u3053\u306e\u5834\u5408\u3067\u3042\u3063\u3066\u3082\u3001[1]\u3092\\(n=5\\)\u306b\u8a2d\u5b9a\u3059\u308b\u3053\u3068\u3067\u8a3c\u660e\u3067\u304d\u307e\u3059\u3002<\/p>\n<h3>\\(n=k+2\\)\u3092\u793a\u3057\u3066\u8a3c\u660e\u3092\u884c\u3046\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5<\/h3>\n<p>\u3053\u3053\u307e\u3067\u3001\u57fa\u672c\u7684\u306a\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u3066\u304d\u307e\u3057\u305f\u3002\u305f\u3060\u5834\u5408\u306b\u3088\u3063\u3066\u306f\u3001\\(n=k+1\\)\u3092\u793a\u3059\u3060\u3051\u3067\u306f\u8a3c\u660e\u3067\u304d\u306a\u3044\u30b1\u30fc\u30b9\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u3001\\(n=k+1\\)\u3060\u3051\u3067\u306a\u304f\u3001\\(n=k+2\\)\u306e\u5834\u5408\u3082\u793a\u3057\u307e\u3057\u3087\u3046\u3002\u3064\u307e\u308a<span style=\"color: #0000ff;\">\\(n=k+1\\)\u306b\u52a0\u3048\u3066\u3001\\(n=k+2\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b\u306e\u3067\u3059\u3002<\/span><\/p>\n<p>\u65b9\u6cd5\u306f\u4ee5\u4e0b\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(n=1\\)\u3068\\(n=2\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u8a3c\u660e\u3059\u308b\u3002<\/li>\n<li>\\(n=k\\)\u3068\\(n=k+1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3001\\(n=k+2\\)\u3067\u3042\u3063\u3066\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b\u3002<\/li>\n<\/ul>\n<p>\u3053\u306e\u5834\u5408\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(n=1\\)\u3068\\(n=2\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064<\/li>\n<li>\\(n=1\\)\u3067\u6210\u308a\u7acb\u3064\u305f\u3081\u3001\\(n=3\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064<\/li>\n<li>\\(n=2\\)\u3067\u6210\u308a\u7acb\u3064\u305f\u3081\u3001\\(n=4\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064<\/li>\n<li>\\(n=3\\)\u3067\u6210\u308a\u7acb\u3064\u305f\u3081\u3001\\(n=5\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064<br \/>\n\u2026\u2026<\/li>\n<\/ul>\n<p>\u3053\u3046\u3057\u3066\u3001\u3059\u3079\u3066\u306e\u81ea\u7136\u6570\\(n\\)\u3067\u6210\u308a\u7acb\u3064\u3068\u8a3c\u660e\u3067\u304d\u307e\u3059\u3002\u305d\u308c\u3067\u306f\u3001\u4ee5\u4e0b\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(x\\)\u3068\\(y\\)\u306b\u3064\u3044\u3066\u3001\\(x+y\\)\u3068\\(xy\\)\u304c\u6574\u6570\u306e\u5834\u5408\u3001\\(x^n+y^n\\)\u304c\u6574\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002\u306a\u304a\u3001\\(n\\)\u306f\u81ea\u7136\u6570\u3067\u3059\u3002<\/li>\n<\/ul>\n<p>\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u8a3c\u660e\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\\(x^k+y^k\\)\u3092\u7528\u3044\u3066\\(x^{k+1}+y^{k+1}\\)\u3092\u8868\u3059\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(x^{k+1}+y^{k+1}=(x^k+y^k)(x+y)\\)\\(-(x^{k-1}+y^{k-1})xy\\)<\/li>\n<\/ul>\n<p>\u305f\u3060\\(x^{k-1}+y^{k-1}\\)\u304c\u6574\u6570\u304b\u3069\u3046\u304b\u308f\u304b\u3089\u306a\u3044\u305f\u3081\u3001\u8a3c\u660e\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\u3002\u305d\u3053\u3067\u4e00\u3064\u524d\u3060\u3051\u3067\u306a\u304f\u3001\u4e8c\u3064\u524d\u3082\u4eee\u5b9a\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u306e\u3067\u3059\u3002\u5b9f\u969b\u306e\u8a3c\u660e\u6cd5\u306f\u4ee5\u4e0b\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p><strong>\u30fb[1] \\(n=1\\)\u3068\\(n=2\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u8a3c\u660e\u3059\u308b<\/strong><\/p>\n<p>\\(n=1\\)\u306e\u3068\u304d\u3001\\(x^1+y^1=x+y\\)\u3067\u3042\u308b\u305f\u3081\u6574\u6570\u3067\u3059\u3002\u307e\u305f\\(n=2\\)\u306e\u3068\u304d\u3001\\(x^2+y^2\\)\\(=(x+y)-2xy\\)\u3067\u3042\u308b\u305f\u3081\u6574\u6570\u3067\u3059\u3002<\/p>\n<p><strong>\u30fb[2] \\(n=k\\)\u3068\\(n=k+1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3001\\(n=k+2\\)\u3067\u3042\u3063\u3066\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b<\/strong><\/p>\n<p>\\(n=k\\)\u3068\\(n=k+1\\)\u306e\u3068\u304d\u306b\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3059\u308b\u3068\u3001\\(x^k+y^k\\)\u3068\\(x^{k+1}+y^{k+1}\\)\u306f\u6574\u6570\u3067\u3059\u3002<\/p>\n<p>\u306a\u304a\u3001\u4eee\u5b9a\u3059\u308b\u3068\u304d\u306b\\(n=k-1\\)\u3068\\(n=k\\)\u3092\u7528\u3044\u308b\u3068\u3001\\(n\\)\u306f\u81ea\u7136\u6570\u3067\u3042\u308b\u305f\u3081\u3001\\(k-1\u22671\\)\u3068\u306a\u308a\u307e\u3059\u3002\u3064\u307e\u308a\\(k\u22672\\)\u3068\u306a\u308b\u305f\u3081\u3001\u3053\u308c\u3092\u907f\u3051\u308b\u305f\u3081\u306b\\(n=k\\)\u3001\\(n=k+1\\)\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002<\/p>\n<p>\u6b21\u306b\u3001\\(n=k+2\\)\u306e\u3068\u304d\u3092\u8003\u3048\u307e\u3057\u3087\u3046\u3002\u305d\u3046\u3059\u308b\u3068\u3001\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(x^{k+2}+y^{k+2}=(x^{k+1}+y^{k+1})(x+y)\\)\\(-(x^k+y^k)xy\\)<\/li>\n<\/ul>\n<p>\\(x+y\\)\u3068\\(xy\\)\u306f\u6574\u6570\u3067\u3042\u308a\u3001\u4eee\u5b9a\u3088\u308a\\(x^k+y^k\\)\u3068\\(x^{k+1}+y^{k+1}\\)\u3082\u6574\u6570\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\\(x^{k+2}+y^{k+2}\\)\u306f\u6574\u6570\u3067\u3059\u3002[1]\u3068[2]\u3088\u308a\u3001\u3059\u3079\u3066\u306e\u81ea\u7136\u6570\\(n\\)\u3067\\(x^{k+2}+y^{k+2}\\)\u306f\u6574\u6570\u3067\u3059\u3002<\/p>\n<h2>\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u5229\u7528\u3057\u3066\u8a3c\u660e\u3059\u308b<\/h2>\n<p>\u6570\u5217\u3084\u6f38\u5316\u5f0f\u306e\u77e5\u8b58\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u8a3c\u660e\u3059\u308b\u65b9\u6cd5\u304c\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u3059\u3002\u6700\u521d\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3057\u305f\u5f8c\u3001\\(n=k+1\\)\u304c\u6210\u308a\u7acb\u3064\u3068\u8a3c\u660e\u3059\u308c\u3070\u3001\u30c9\u30df\u30ce\u5012\u3057\u3068\u540c\u69d8\u306b\u3059\u3079\u3066\u3092\u8a3c\u660e\u3067\u304d\u307e\u3059\u3002\u3053\u308c\u304c\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306e\u539f\u7406\u3067\u3059\u3002<\/p>\n<p>\u57fa\u672c\u3068\u306a\u308b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u306f\u3001\\(n=1\\)\u3068\\(n=k+1\\)\u3092\u8a3c\u660e\u3059\u308c\u3070\u3044\u3044\u3067\u3059\u3002\u305f\u3060\u5fdc\u7528\u554f\u984c\u3067\u306f\u3001\u307b\u304b\u306e\u65b9\u6cd5\u3092\u5229\u7528\u3057\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n<p>\u4f8b\u3048\u3070\\(a&gt;b\\)\u3092\u8a3c\u660e\u3057\u305f\u3044\u5834\u5408\u3001\u4e21\u8fba\u306e\u5dee\u3092\u5229\u7528\u3057\u3066\u3001\\(a-b&gt;0\\)\u3092\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002\u307e\u305f\\(n=k+1\\)\u3092\u7528\u3044\u3066\u3082\u8a3c\u660e\u3067\u304d\u306a\u3044\u5834\u5408\u3001\\(n=k+2\\)\u3082\u5229\u7528\u3057\u3066\u3001\u4e00\u3064\u524d\u3060\u3051\u3067\u306a\u304f\u4e8c\u3064\u524d\u3082\u4eee\u5b9a\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u5b9f\u969b\u306b\u4f8b\u984c\u3092\u89e3\u304b\u306a\u3044\u3068\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u5229\u7528\u3057\u3066\u306e\u8a3c\u660e\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u305d\u3053\u3067\u554f\u984c\u3092\u89e3\u304d\u3001\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u5229\u7528\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u3087\u3046\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6570\u5217\u304c\u95a2\u308f\u308b\u8a3c\u660e\u6cd5\u306b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u304c\u3042\u308a\u307e\u3059\u3002\\(n=1\\)\u3068\\(n=k+1\\)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u3059\u3079\u3066\u306e\u6761\u4ef6\u304c\u6210\u7acb\u3059\u308b\u3068\u8a3c\u660e\u3067\u304d\u308b\u65b9\u6cd5\u304c\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u3059\u3002 \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u5229\u7528\u3057\u3066\u8a3c\u660e\u3059\u308b\u3068\u304d\u3001\u5b9f\u969b\u306b\u554f\u984c [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":12235,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":{"0":"post-12224","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-h-math"},"_links":{"self":[{"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/posts\/12224","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/comments?post=12224"}],"version-history":[{"count":13,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/posts\/12224\/revisions"}],"predecessor-version":[{"id":12353,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/posts\/12224\/revisions\/12353"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/media\/12235"}],"wp:attachment":[{"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/media?parent=12224"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/categories?post=12224"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/tags?post=12224"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}