{"id":11953,"date":"2022-12-29T21:52:57","date_gmt":"2022-12-29T12:52:57","guid":{"rendered":"https:\/\/hatsudy.com\/jp\/?p=11953"},"modified":"2023-01-05T09:43:53","modified_gmt":"2023-01-05T00:43:53","slug":"sequence-3","status":"publish","type":"post","link":"https:\/\/hatsudy.com\/jp\/sequence-3.html","title":{"rendered":"\u30b7\u30b0\u30de\u03a3\u306e\u8a08\u7b97\u3068\u516c\u5f0f\uff1a\u4e00\u822c\u9805\u3068\u7b49\u5dee\u6570\u5217\u30fb\u7b49\u6bd4\u6570\u5217\u306e\u548c"},"content":{"rendered":"\n<p>\u6570\u5217\u3067\u5b66\u3076\u91cd\u8981\u306a\u8a18\u53f7\u306b\u30b7\u30b0\u30de\uff08\\(\u03a3\\)\uff09\u304c\u3042\u308a\u307e\u3059\u3002\u8a08\u7b97\u3067\u3072\u3093\u3071\u3093\u306b\u5229\u7528\u3055\u308c\u308b\u8a18\u53f7\u304c\u30b7\u30b0\u30de\u3067\u3042\u308a\u3001\u516c\u5f0f\u3092\u5fc5\u305a\u899a\u3048\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002<\/p>\n<p>\u7b49\u5dee\u6570\u5217\u3084\u7b49\u6bd4\u6570\u5217\u306e\u548c\u3092\u8a08\u7b97\u3059\u308b\u3068\u304d\u3001\u30b7\u30b0\u30de\u8a18\u53f7\u304c\u5229\u7528\u3055\u308c\u307e\u3059\u3002\u305d\u308c\u305e\u308c\u306e\u6570\u5217\u3092\u8a08\u7b97\u3059\u308b\u5834\u5408\u3001\u591a\u304f\u306e\u6642\u9593\u304c\u304b\u304b\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u5f0f\u3092\u307e\u3068\u3081\u305f\u3042\u3068\u3001\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u5229\u7528\u3057\u3066\u8a08\u7b97\u3059\u308c\u3070\u5927\u5e45\u306a\u6642\u9593\u77ed\u7e2e\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u5229\u7528\u3059\u308b\u305f\u3081\u306b\u306f\u3001\u4e00\u822c\u9805\u3092\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u305d\u306e\u5f8c\u3001\u516c\u5f0f\u3092\u5229\u7528\u3057\u3066\u548c\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u5229\u7528\u3059\u308b\u3068\u304d\u306e\u516c\u5f0f\u3084\u6027\u8cea\u306b\u306f\u4f55\u304c\u3042\u308b\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u6570\u5217\u3067\u91cd\u8981\u306a\u30b7\u30b0\u30de\u8a18\u53f7\u306e\u8a08\u7b97\u65b9\u6cd5\u3092\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<h2>\u6570\u5217\u3067\u548c\u3092\u8a08\u7b97\u3059\u308b\u30b7\u30b0\u30de\u8a18\u53f7\uff08\\(\u03a3\\)\uff09\u3068\u610f\u5473<\/h2>\n<p>\u6570\u5217\u3067\u548c\u3092\u8a08\u7b97\u3057\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u5834\u9762\u306f\u591a\u3044\u3067\u3059\u3002\u305f\u3060\u3001\u6570\u5217\u3067\u306f\u975e\u5e38\u306b\u591a\u304f\u306e\u6570\u5b57\u3092\u7f85\u5217\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u4f8b\u3048\u3070\u3001\u6570\u5217\u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u6570\u5b57\u304c\u9023\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(x_1+x_2+&#8230;+x_n\\)<\/li>\n<\/ul>\n<p>\u305d\u3053\u3067\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\\(\u03a3\\)\u3092\u7528\u3044\u3066\u8868\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(x_1+x_2+&#8230;+x_n=\\displaystyle\\sum_{k=1}^{n}{x_k}\\)<\/li>\n<\/ul>\n<p>\u30b7\u30b0\u30de\u3092\u5229\u7528\u3059\u308c\u3070\u3001\u3053\u306e\u3088\u3046\u306b\u7c21\u6f54\u306b\u8a18\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<span style=\"color: #0000ff;\">\u30b7\u30b0\u30de\u3068\u3044\u3046\u306e\u306f\u3001\\(k\\)\u306b1\u304b\u3089\\(n\\)\u307e\u3067\u4ee3\u5165\u3057\u3001\u3059\u3079\u3066\u8db3\u3059\u3053\u3068\u3092\u610f\u5473\u3057\u3066\u3044\u307e\u3059\u3002<\/span>\u4f8b\u3048\u3070\\(\\displaystyle\\sum_{k=1}^{5}{2k}\\)\u306f\u4ee5\u4e0b\u306e\u5f0f\u3092\u610f\u5473\u3057\u307e\u3059\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/hatsudy.com\/jp\/wp-content\/uploads\/2022\/12\/dhw2.jpg\" alt=\"\" width=\"600\" height=\"111\" class=\"aligncenter size-full wp-image-11966\" \/><\/p>\n<p>\u3053\u3046\u3057\u305f\u5177\u4f53\u4f8b\u3092\u53c2\u8003\u306b\u3057\u3066\u3001\u30b7\u30b0\u30de\u304c\u4f55\u3092\u610f\u5473\u3057\u3066\u3044\u308b\u306e\u304b\u7406\u89e3\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<h3>\u30b7\u30b0\u30de\u306e\u8a08\u7b97\u3067\u5fc5\u9808\u3068\u306a\u308b5\u3064\u306e\u516c\u5f0f<\/h3>\n<p>\u30b7\u30b0\u30de\u306e\u8a08\u7b97\u3092\u3059\u308b\u3068\u304d\u3001\u3042\u306a\u305f\u304c\u899a\u3048\u308b\u3079\u304d\u516c\u5f0f\u306f5\u3064\u3042\u308a\u307e\u3059\u3002\u305f\u3060\u3001\u3053\u308c\u3089\u306e\u516c\u5f0f\u306e\u3046\u30613\u3064\u306f\u3059\u3067\u306b\u5b66\u3093\u3067\u3044\u308b\u305f\u3081\u3001\u65b0\u305f\u306b\u899a\u3048\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u516c\u5f0f\u306f2\u3064\uff082\u4e57\u306e\u548c\u30683\u4e57\u306e\u548c\uff09\u3067\u3059\u3002\u305d\u308c\u3067\u306f\u3001\u4ee5\u4e0b\u306b\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u5229\u7528\u3059\u308b\u516c\u5f0f\u3092\u8a18\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<p><strong>\u30fb\\(a\\)\u304c\\(n\\)\u500b\u3042\u308b\u6570\u5217<\/strong><\/p>\n<p>\\(a\\)\u304c\\(n\\)\u500b\u4e26\u3093\u3067\u3044\u308b\u5834\u5408\u3001\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\sum_{k=1}^{n}{a}=a+a+&#8230;+a\\)<\/li>\n<\/ul>\n<p>\u3053\u306e\u5834\u5408\u3001\\(\\displaystyle\\sum_{k=1}^{n}{a}=an\\)\u3067\u3059\u3002\u8aac\u660e\u3057\u306a\u304f\u3066\u3082\u3001\u3053\u306e\u516c\u5f0f\u304c\u6210\u308a\u7acb\u3064\u7406\u7531\u3092\u7406\u89e3\u3067\u304d\u308b\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n<p><strong>\u30fb1\u304b\u3089\\(n\\)\u307e\u3067\u306e\u548c<\/strong><\/p>\n<p>\u6570\u5217\u3067\u3072\u3093\u3071\u3093\u306b\u5229\u7528\u3055\u308c\u308b\u306e\u304c\u30011\u304b\u3089\\(n\\)\u307e\u3067\u306e\u548c\u3092\u610f\u5473\u3057\u307e\u3059\u3002\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u5229\u7528\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\sum_{k=1}^{n}{k}=1+2+&#8230;+n\\)<\/li>\n<\/ul>\n<p>1\u304b\u3089\\(n\\)\u307e\u3067\u306e\u548c\u3068\u3044\u3046\u306e\u306f\u3001\u521d\u98051\u3001\u672b\u9805\\(n\\)\u306e\u7b49\u5dee\u6570\u5217\u3067\u3059\u3002\u305d\u3053\u3067\u7b49\u5dee\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f\u306b\u5bfe\u3057\u3066\u3001\u521d\u98051\u3001\u672b\u9805\\(n\\)\u3092\u4ee3\u5165\u3057\u307e\u3057\u3087\u3046\u3002\u305d\u3046\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u516c\u5f0f\u3092\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\sum_{k=1}^{n}{k}=\\displaystyle\\frac{1}{2}n(n+1)\\)<\/li>\n<\/ul>\n<p>\u65b0\u305f\u306a\u516c\u5f0f\u3068\u3057\u3066\u899a\u3048\u3066\u306f\u3044\u3051\u307e\u305b\u3093\u3002\u7b49\u5dee\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f\u3092\u5229\u7528\u3059\u308c\u3070\u30011\u304b\u3089\\(n\\)\u307e\u3067\u306e\u548c\u3092\u5c0e\u51fa\u3067\u304d\u307e\u3059\u3002<\/p>\n<p><strong>\u30fb2\u4e57\u306e\u548c<\/strong><\/p>\n<p>\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u5b66\u3076\u3068\u304d\u3001\u65b0\u305f\u306b\u899a\u3048\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u5185\u5bb9\u304c2\u4e57\u306e\u548c\u3067\u3059\u30022\u4e57\u306e\u548c\u306f\u3072\u3093\u3071\u3093\u306b\u5229\u7528\u3055\u308c\u308b\u305f\u3081\u3001\u516c\u5f0f\u3092\u899a\u3048\u3066\u3044\u306a\u3044\u3068\u8a08\u7b97\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\u3002\u4e8c\u4e57\u306e\u548c\u3068\u3044\u3046\u306e\u306f\u3001\u4ee5\u4e0b\u306e\u8db3\u3057\u7b97\u3092\u6307\u3057\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\sum_{k=1}^{n}{k^2}=1^2+2^2+3^2+&#8230;+n^2\\)<\/li>\n<\/ul>\n<p>\u8a73\u7d30\uff08\u516c\u5f0f\u306e\u8a3c\u660e\uff09\u306f\u7701\u304d\u307e\u3059\u304c\u3001\u4e8c\u4e57\u306e\u548c\u306e\u516c\u5f0f\u306f\u4ee5\u4e0b\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\sum_{k=1}^{n}{k^2}=\\displaystyle\\frac{1}{6}n(n+1)(2n+1)\\)<\/li>\n<\/ul>\n<p>\u3053\u306e\u516c\u5f0f\u3092\u899a\u3048\u3001\u8db3\u3057\u7b97\u3092\u884c\u3048\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u3087\u3046\u3002<\/p>\n<p><strong>\u30fb3\u4e57\u306e\u548c<\/strong><\/p>\n<p>2\u4e57\u306e\u548c\u307b\u3069\u91cd\u8981\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u304c\u30013\u4e57\u306e\u548c\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002\u4ee5\u4e0b\u306e\u8db3\u3057\u7b97\u304c3\u4e57\u306e\u548c\u3067\u3059\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\sum_{k=1}^{n}{k^3}=1^3+2^3+3^3+&#8230;+n^3\\)<\/li>\n<\/ul>\n<p>3\u4e57\u306e\u548c\u306e\u516c\u5f0f\u306f\u4ee5\u4e0b\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\sum_{k=1}^{n}{k^3}=\\left\\{\\displaystyle\\frac{1}{2}n(n+1)\\right\\}^2\\)<\/li>\n<\/ul>\n<p>\u5148\u307b\u3069\u30011\u304b\u3089\\(n\\)\u307e\u3067\u306e\u548c\u304c\\(\\displaystyle\\sum_{k=1}^{n}{k}=\\displaystyle\\frac{1}{2}n(n+1)\\)\u306b\u306a\u308b\u3068\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002\u3053\u306e\u516c\u5f0f\u3092\u4e8c\u4e57\u3059\u308b\u3068\u30013\u4e57\u306e\u548c\u306b\u306a\u308a\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u516c\u5f0f\u306f\u899a\u3048\u3084\u3059\u3044\u3067\u3059\u3002<\/p>\n<p><strong>\u30fb\u7b49\u6bd4\u6570\u5217\u3067\u306e\u7b2c\\(n\\)\u9805\u307e\u3067\u306e\u548c<\/strong><\/p>\n<p>\u7b49\u6bd4\u6570\u5217\u3092\u5b66\u3076\u3068\u304d\u3001\u7b49\u6bd4\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f\u3092\u5fc5\u305a\u7fd2\u3044\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u7b49\u6bd4\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f\u306f\u3059\u3067\u306b\u899a\u3048\u3066\u3044\u308b\u3068\u601d\u3044\u307e\u3059\u3002\u521d\u9805\\(a\\)\u3001\u516c\u6bd4\\(r\\)\u3001\u9805\u6570\\(n\\)\u306e\u7b49\u6bd4\u6570\u5217\u306e\u4e00\u822c\u9805\u306f\\(a_n=ar^{n-1}\\)\u3067\u3059\u3002<\/p>\n<p>\u3053\u306e\u3068\u304d\u3001\u7b49\u6bd4\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f\u306f\u30b7\u30b0\u30de\u8a18\u53f7\u3068\u7b49\u6bd4\u6570\u5217\u306e\u4e00\u822c\u9805\u3092\u5229\u7528\u3057\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\sum_{k=1}^{n}{ar^{k-1}}=\\displaystyle\\frac{a(r^n-1)}{r-1}\\)<\/li>\n<\/ul>\n<p>\u3059\u3067\u306b\u77e5\u3063\u3066\u3044\u308b\u5185\u5bb9\u3067\u3042\u308b\u305f\u3081\u3001\u65b0\u305f\u306b\u899a\u3048\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u516c\u5f0f\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n<h3>\u30b7\u30b0\u30de\u8a18\u53f7\u306e\u6027\u8cea\uff1a\u8db3\u3057\u7b97\u3084\u5f15\u304d\u7b97\u3001\u5b9f\u6570\u500d<\/h3>\n<p>\u305d\u308c\u3067\u306f\u3001\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u4f7f\u7528\u3059\u308b\u3068\u304d\u306e\u6027\u8cea\u3092\u5b66\u3073\u307e\u3057\u3087\u3046\u3002\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u542b\u3080\u5f0f\u306b\u3064\u3044\u3066\u3001\u8db3\u3057\u7b97\u3084\u5f15\u304d\u7b97\u3001\u5b9f\u6570\u500d\u3092\u3059\u308b\u3068\u304d\u306e\u65b9\u6cd5\u3092\u7406\u89e3\u3059\u308b\u306e\u3067\u3059\u3002<\/p>\n<p>\u307e\u305a\u3001\u6570\u5217\u306e\u8db3\u3057\u7b97\uff08\u307e\u305f\u306f\u5f15\u304d\u7b97\uff09\u3092\u5b66\u3073\u307e\u3057\u3087\u3046\u3002\u6570\u5217\\(\\{a_n\\}\\)\u306e\u548c\u3068\u6570\u5217\\(\\{b_n\\}\\)\u306e\u548c\u306b\u306f\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\sum_{k=1}^{n}{(a_k+b_k)}=\\displaystyle\\sum_{k=1}^{n}{a_k}\\)\\(+\\displaystyle\\sum_{k=1}^{n}{b_k}\\)<\/li>\n<\/ul>\n<p>\u3064\u307e\u308a\u3001<span style=\"color: #0000ff;\">\u305d\u308c\u305e\u308c\u306e\u6570\u5217\u306e\u548c\u3092\u8a08\u7b97\u3059\u308b\u3068\u304d\u3001\u5148\u306b\u5f0f\u306e\u8db3\u3057\u7b97\u3084\u5f15\u304d\u7b97\u3092\u3057\u305f\u5f8c\u3001\u30b7\u30b0\u30de\u3092\u5229\u7528\u3057\u3066\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/span><\/p>\n<p>\u307e\u305f\u5b9a\u6570\\(p\\)\u306b\u3064\u3044\u3066\u3001\u6570\u5217\u306e\u548c\u306e\u8a08\u7b97\u3067\u306f\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\sum_{k=1}^{n}{pa_k}=p\\displaystyle\\sum_{k=1}^{n}{a_k}\\)<\/li>\n<\/ul>\n<p>\u3053\u306e\u3088\u3046\u306b\u3001<span style=\"color: #0000ff;\">\u5b9a\u6570\u3092\u30b7\u30b0\u30de\u306e\u5916\u306b\u51fa\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/span>\u3053\u308c\u3089\u306e\u6027\u8cea\u3092\u5229\u7528\u3057\u3066\u3001\u30b7\u30b0\u30de\u3092\u5229\u7528\u3059\u308b\u8a08\u7b97\u554f\u984c\u3092\u89e3\u3051\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u3087\u3046\u3002<\/p>\n<h2>\u4e00\u822c\u9805\u3092\u6c42\u3081\u305f\u5f8c\u3001\u30b7\u30b0\u30de\u3092\u5229\u7528\u3057\u3066\u8a08\u7b97\u3059\u308b<\/h2>\n<p>\u305d\u308c\u3067\u306f\u3001\u5b9f\u969b\u306b\u8a08\u7b97\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u306b\u3088\u3063\u3066\u6570\u5217\u306e\u548c\u306e\u8a08\u7b97\u306b\u6163\u308c\u307e\u3057\u3087\u3046\u3002<span style=\"color: #ff0000;\">\u6570\u5217\u306e\u548c\u306e\u8a08\u7b97\u3092\u3059\u308b\u3068\u304d\u3001\u307e\u305a\u4e00\u822c\u9805\u3092\u5f97\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/span>\u305d\u306e\u5f8c\u3001\u516c\u5f0f\u3092\u5229\u7528\u3057\u3066\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002\u305d\u308c\u3067\u306f\u3001\u4ee5\u4e0b\u306e\u6570\u5217\u306e\u548c\u306f\u4f55\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<ul>\n<li>\\(1,1+2,1+2+2^2,\u2026\\)<\/li>\n<\/ul>\n<p>\u4e00\u822c\u9805\u3092\\(a_n\\)\u3068\u3059\u308b\u3068\u3001\u4e00\u822c\u9805\u306f\u521d\u98051\u3001\u516c\u6bd42\u3068\u3059\u308b\u7b49\u6bd4\u6570\u5217\u306e\u548c\u3067\u3042\u308b\u3068\u308f\u304b\u308a\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u4e00\u822c\u9805\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(a_n=\\displaystyle\\frac{1\u00b7(2^n-1)}{2-1}\\)<\/p>\n<p>\\(a_n=2^n-1\\)<\/p>\n<p>\u305d\u3053\u3067\u3001\\(a_n\\)\u306e\u548c\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(\\displaystyle\\sum_{k=1}^{n}{(2^k-1)}\\)<\/p>\n<p>\\(=\\displaystyle\\sum_{k=1}^{n}{2\u00b72^{k-1}}\\)\\(-\\displaystyle\\sum_{k=1}^{n}{1}\\)<\/p>\n<p>\\(=\\displaystyle\\frac{2(2^n-1)}{2-1}-n\\)<\/p>\n<p>\\(=2^{n+1}-n-2\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\\(a_n\\)\u306e\u548c\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<h3>\u7b2c\\(k\\)\u9805\u306b\\(n\\)\u3092\u542b\u3080\u6570\u5217\u306e\u548c<\/h3>\n<p>\u306a\u304a\u3001\u5fdc\u7528\u554f\u984c\u3092\u89e3\u3051\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u3087\u3046\u3002\u5177\u4f53\u7684\u306b\u306f\u3001\u7b2c\\(k\\)\u9805\u306b\\(n\\)\u3092\u542b\u3080\u5834\u5408\u306e\u6570\u5217\u306e\u548c\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002\u8a00\u3044\u63db\u3048\u308b\u3068\u3001\u521d\u9805\u306b\\(n\\)\u3092\u542b\u3080\u6570\u5217\u306e\u548c\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002\u4ee5\u4e0b\u306e\u6570\u5217\u306b\u3064\u3044\u3066\u3001\u548c\\(S\\)\u306f\u3044\u304f\u3089\u306b\u306a\u308b\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<ul>\n<li>\\(S_n=1\u00b7n+2\u00b7(n-1)+3\u00b7(n-2)\\)\\(+&#8230;\\)\\(+(n-1)\u00b72\\)\\(+n\u00b71\\)<\/li>\n<\/ul>\n<p>\u3053\u306e\u8a08\u7b97\u3092\u3059\u308b\u305f\u3081\u3001\u6570\u5217\u306e\u5de6\u5074\u3068\u53f3\u5074\u3067\u5206\u3051\u3066\u8003\u3048\u307e\u3057\u3087\u3046\u3002\u305d\u3046\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p><strong>\u30fb\u6570\u5217\u306e\u5de6\u5074<\/strong><\/p>\n<p>\\(1+2+3+&#8230;+n-1+n\\)<\/p>\n<p>\u3053\u308c\u3092\u4e00\u822c\u5f0f\u306b\u3059\u308b\u3068\u3001\\(a_n=n\\)\u306b\u306a\u308a\u307e\u3059\u3002\u307e\u305f\u3001\u7b2c\\(k\\)\u9805\u306e\u5024\u306f\\(k\\)\u3067\u3059\u3002<\/p>\n<p><strong>\u30fb\u6570\u5217\u306e\u53f3\u5074<\/strong><\/p>\n<p>\\(n+(n-1)+(n-2)+&#8230;+2+1\\)<\/p>\n<p>\u3053\u306e\u6570\u5217\u306b\u3064\u3044\u3066\u306f\u3001\u521d\u9805\\(n\\)\u3001\u516c\u5dee\\(-1\\)\u3001\u672b\u98051\u3001\u9805\u6570\\(n\\)\u306e\u7b49\u5dee\u6570\u5217\u3068\u308f\u304b\u308a\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u7b2c\\(k\\)\u9805\u306e\u5024\u306f\u4f55\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<p>\u6ce8\u610f\u70b9\u3068\u3057\u3066\u3001\u7b2c\\(k\\)\u9805\u306e\u5024\u306f\\(k+(k-1)\u00b7(-1)\\)\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<span style=\"color: #0000ff;\">\u9805\u6570\u306f\\(k\\)\u3068\u9023\u52d5\u3059\u308b\u3082\u306e\u306e\u3001\u521d\u9805\\(n\\)\u306f\\(k\\)\u3068\u9023\u52d5\u3057\u307e\u305b\u3093\u3002<\/span>\u305d\u306e\u305f\u3081\u7b2c\\(k\\)\u9805\u306e\u5f0f\u3092\u8a18\u3059\u3068\u304d\u3001\u521d\u9805\\(n\\)\u3092\\(k\\)\u306b\u5909\u3048\u3066\u306f\u3044\u3051\u307e\u305b\u3093\u3002\u9805\u6570\u306e\u5024\u304c\u4f55\u3067\u3042\u3063\u3066\u3082\u3001\u521d\u9805\u306f\\(n\\)\u3067\u3042\u308b\u305f\u3081\u3001\u7b2c\\(k\\)\u9805\u306e\u5024\u306f\\(n+(k-1)\u00b7(-1)\\)\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u4f8b\u3048\u3070\u7b2c\u4e00\u9805\u306e\u5834\u5408\u3001\\(k=1\\)\u3092\u4ee3\u5165\u3057\u307e\u3057\u3087\u3046\u3002\u5f0f\u304c\\(k+(k-1)\u00b7(-1)\\)\u306e\u5834\u5408\u3001\\(k=1\\)\u3060\u3068\u3001\u5024\u306f1\u306b\u306a\u308a\u307e\u3059\u3002\u7b2c\u4e00\u9805\uff08\u521d\u9805\uff09\u306f\\(n\\)\u3067\u3042\u308b\u305f\u3081\u3001\u660e\u3089\u304b\u306b\u9593\u9055\u3063\u3066\u3044\u307e\u3059\u3002\u4e00\u65b9\u3067\\(n+(k-1)\u00b7(-1)\\)\u306b\u5bfe\u3057\u3066\\(k=1\\)\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001\u7b54\u3048\u306f\\(n\\)\u306b\u306a\u308a\u3001\u521d\u9805\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002<\/p>\n<p>\u521d\u9805\u306b\\(n\\)\u3092\u542b\u3080\u5834\u5408\u3001\u7b2c\\(k\\)\u9805\u306b\\(n\\)\u3092\u542b\u307f\u307e\u3059\u3002\u521d\u9805\u306b\u542b\u307e\u308c\u308b\\(n\\)\u3068\u9805\u6570\\(n\\)\u306f\u7570\u306a\u308b\u6982\u5ff5\u3067\u3042\u308b\u3053\u3068\u3092\u8a8d\u8b58\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u306a\u304a\u3001\u5f0f\u3092\u30b7\u30f3\u30d7\u30eb\u306b\u3059\u308b\u305f\u3081\u306b\u4ee5\u4e0b\u306e\u8a08\u7b97\u3092\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(n+(k-1)\u00b7(-1)\\)<\/p>\n<p>\\(n-k+1\\)<\/p>\n<p><strong>\u30fb\u7b2c\\(k\\)\u9805\u3067\u306e\u4e00\u822c\u9805<\/strong><\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u6570\u5217\u306e\u5de6\u5074\u306e\u4e00\u822c\u9805\u3068\u53f3\u5074\u306e\u4e00\u822c\u9805\u3092\u5229\u7528\u3057\u3066\u3001\u554f\u984c\u6587\u306e\u5f0f\u3067\u306e\u4e00\u822c\u9805\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002\u7b2c\\(k\\)\u9805\u306e\u5024\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(k(n-k+1)\\)<\/p>\n<p>\\(=-k^2+kn+k\\)<\/p>\n<p>\\(=-k^2+k(n+1)\\)<\/p>\n<p>\u3053\u306e\u4e00\u822c\u9805\u3092\u3082\u3064\u6570\u5217\u306e\u548c\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(\\displaystyle\\sum_{k=1}^{n}{\\{-k^2+k(n+1)\\}}\\)<\/p>\n<p>\\(=-\\displaystyle\\frac{1}{6}n(n+1)(2n+1)\\)\\(+\\displaystyle\\frac{1}{2}n(n+1)\u00d7(n+1)\\)<\/p>\n<p>\\(=\\displaystyle\\frac{1}{6}n(n+1)\\)\\(\\{-(2n+1)\\)\\(+3(n+1)\\}\\)<\/p>\n<p>\\(=\\displaystyle\\frac{1}{6}n(n+1)(n+2)\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u7b2c\\(k\\)\u9805\u306b\\(n\\)\u3092\u542b\u3080\u5834\u5408\uff08\u521d\u9805\u306b\\(n\\)\u3092\u542b\u3080\u5834\u5408\uff09\u3001\u521d\u9805\u3068\u9805\u6570\u3092\u533a\u5225\u3057\u3066\u5f0f\u3092\u4f5c\u308a\u307e\u3057\u3087\u3046\u3002<\/p>\n<h2>\u30b7\u30b0\u30de\u3092\u5229\u7528\u3057\u3066\u6570\u5217\u306e\u548c\u3092\u8a08\u7b97\u3059\u308b<\/h2>\n<p>\u6570\u5217\u3067\u548c\u3092\u8a08\u7b97\u3057\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u3053\u3068\u306f\u591a\u3044\u3067\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u30b7\u30b0\u30de\u3092\u5229\u7528\u3059\u308c\u3070\u8a18\u8f09\u3059\u308b\u3079\u304d\u5f0f\u304c\u5c11\u306a\u304f\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u307e\u305f\u3001\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u5229\u7528\u3059\u308b\u3068\u304d\u306f\u516c\u5f0f\u3092\u4f7f\u3048\u307e\u3059\u3002\u516c\u5f0f\u3092\u899a\u3048\u308b\u5fc5\u8981\u306f\u3042\u308a\u307e\u3059\u304c\u3001\u516c\u5f0f\u3092\u5229\u7528\u3059\u308c\u3070\u8a08\u7b97\u304c\u697d\u3067\u3059\u3002\u306a\u304a5\u3064\u3042\u308b\u516c\u5f0f\u306e\u3046\u3061\u3001\u65b0\u305f\u306b\u899a\u3048\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u516c\u5f0f\u306f2\u4e57\u306e\u548c\u30683\u4e57\u306e\u548c\u306e2\u3064\u3067\u3059\u3002\u65b0\u305f\u306a\u516c\u5f0f\u306e\u4e2d\u3067\u3082\u3001\u7279\u306b2\u4e57\u306e\u548c\u306f\u91cd\u8981\u3067\u3059\u3002<\/p>\n<p>\u306a\u304a\u30b7\u30b0\u30de\u8a18\u53f7\u3092\u5229\u7528\u3057\u3066\u8a08\u7b97\u3059\u308b\u3068\u304d\u3001\u4e00\u822c\u9805\u3092\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u521d\u9805\u306b\\(n\\)\u304c\u542b\u307e\u308c\u308b\u5834\u5408\u3001\u521d\u9805\\(n\\)\u3068\u9805\u6570\\(k\\)\u306e\u9055\u3044\u306b\u6ce8\u610f\u3057\u3066\u3001\u7b2c\\(k\\)\u9805\u306e\u5024\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u9ad8\u6821\u6570\u5b66\u306b\u9650\u3089\u305a\u3001\u5927\u5b66\u7269\u7406\u3084\u7d71\u8a08\u5b66\u3001\u91d1\u878d\uff08\u304a\u91d1\u306e\u8a08\u7b97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