{"id":12075,"date":"2023-01-05T12:51:31","date_gmt":"2023-01-05T03:51:31","guid":{"rendered":"https:\/\/hatsudy.com\/jp\/?p=12075"},"modified":"2023-01-11T09:26:34","modified_gmt":"2023-01-11T00:26:34","slug":"recurrence-relation-2","status":"publish","type":"post","link":"https:\/\/hatsudy.com\/jp\/recurrence-relation-2.html","title":{"rendered":"\u7279\u6027\u65b9\u7a0b\u5f0f\u3068\u7279\u6b8a\u306a\u6f38\u5316\u5f0f\uff1a\u03b1\u3092\u5229\u7528\u3059\u308b\u554f\u984c\u306e\u89e3\u304d\u65b9"},"content":{"rendered":"\n<p>\u6f38\u5316\u5f0f\u306b\u95a2\u3059\u308b\u554f\u984c\u3092\u89e3\u304f\u3068\u304d\u3001\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u304c\u591a\u3044\u3067\u3059\u3002\u03b1\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u6f38\u5316\u5f0f\u3092\u5909\u5f62\u3057\u3066\u6570\u5217\u306e\u4e00\u822c\u5f0f\u3092\u5f97\u308b\u306e\u3067\u3059\u3002<\/p>\n<p>\u305f\u3060\u3001\u591a\u304f\u306e\u4eba\u306f\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5b66\u3076\u3068\u304d\u306b\u6df7\u4e71\u3057\u307e\u3059\u3002\u7570\u306a\u308b\u6570\u5b57\u3092\u03b1\u306b\u7f6e\u304d\u63db\u3048\u308b\u305f\u3081\u3001\u306a\u305c\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3067\u304d\u308b\u306e\u304b\u7406\u89e3\u3067\u304d\u306a\u3044\u306e\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u3053\u306e\u7406\u7531\u3092\u77e5\u3089\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002<\/p>\n<p>\u307e\u305f\u6f38\u5316\u5f0f\u3067\u306f\u3001\u307b\u304b\u306b\u3082\u7279\u6b8a\u306a\u6f38\u5316\u5f0f\u304c\u305f\u304f\u3055\u3093\u3042\u308a\u307e\u3059\u3002\u6f38\u5316\u5f0f\u306e\u89e3\u304d\u65b9\u306f\u6c7a\u307e\u3063\u3066\u3044\u308b\u305f\u3081\u3001\u7b54\u3048\u3092\u5f97\u308b\u305f\u3081\u306b\u306f\u3001\u554f\u984c\u306e\u89e3\u304d\u65b9\u3092\u77e5\u3063\u3066\u3044\u308b\u304b\u3069\u3046\u304b\u304c\u91cd\u8981\u3067\u3059\u3002\u306a\u304a\u554f\u984c\u306e\u89e3\u304d\u65b9\u3092\u77e5\u3089\u306a\u3044\u5834\u5408\u3001\u304a\u305d\u3089\u304f\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u30d2\u30f3\u30c8\u306a\u3057\u3067\u6f38\u5316\u5f0f\u306e\u554f\u984c\u3092\u89e3\u304f\u306e\u306f\u96e3\u3057\u3044\u304b\u3089\u3067\u3059\u3002<\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u3069\u306e\u3088\u3046\u306b\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3059\u308c\u3070\u3044\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u307e\u305f\u3001\u7279\u6b8a\u306a\u6f38\u5316\u5f0f\u306b\u3064\u3044\u3066\u3069\u3046\u7b54\u3048\u3092\u5f97\u308c\u3070\u3044\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u6f38\u5316\u5f0f\u306e\u5fdc\u7528\u554f\u984c\u306e\u89e3\u304d\u65b9\u3092\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<h2>\u7279\u6027\u65b9\u7a0b\u5f0f\uff08\\(a_{n+1}=pa_n+q\\)\uff09\u3092\u5229\u7528\u3059\u308b\u6f38\u5316\u5f0f\u306e\u89e3\u304d\u65b9<\/h2>\n<p>\u6f38\u5316\u5f0f\u3092\u5229\u7528\u3057\u3066\u4e00\u822c\u9805\u3092\u5f97\u308b\u305f\u3081\u306b\u306f\u3001\u5f0f\u3092\u7b49\u5dee\u6570\u5217\u578b\u3001\u7b49\u6bd4\u6570\u5217\u578b\u3001\u968e\u5dee\u6570\u5217\u578b\u306e\u3046\u3061\u3001\u3069\u308c\u304b\u306b\u5909\u5f62\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u308c\u3089\u306e\u6f38\u5316\u5f0f\u306b\u306a\u3063\u3066\u3044\u306a\u3044\u5834\u5408\u3001\u5f0f\u3092\u5909\u5f62\u3057\u307e\u3057\u3087\u3046\u3002\u3053\u306e\u3068\u304d\u5229\u7528\u3059\u308b\u306e\u304c\u7279\u6027\u65b9\u7a0b\u5f0f\u3067\u3059\u3002<\/p>\n<p>\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u7406\u89e3\u3059\u308b\u305f\u3081\u3001\u4f8b\u984c\u3092\u89e3\u3044\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(a_1=1\\)\u3001\\(a_{n+1}=3a_n+4\\)\u306e\u3068\u304d\u3001\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\u3053\u306e\u6f38\u5316\u5f0f\u3067\u306f\u3001\u7b49\u5dee\u6570\u5217\u578b\u306b\u3082\u7b49\u6bd4\u6570\u5217\u578b\u306b\u3082\u306a\u3063\u3066\u3044\u307e\u305b\u3093\u3002\u3064\u307e\u308a\u3001\u3053\u306e\u72b6\u614b\u306e\u307e\u307e\u3067\u306f\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\u3002\u305d\u3053\u3067\u3001<span style=\"color: #ff0000;\">\\(a_{n+1}=\u03b1\\)\u3001\\(a_n=\u03b1\\)\u3068\u3057\u307e\u3057\u3087\u3046\u3002<\/span>\u3053\u308c\u306b\u3088\u308a\u3001\\(\u03b1=3\u03b1+4\\)\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u306e\u5f0f\u3092\u7279\u6027\u65b9\u7a0b\u5f0f\u3068\u3044\u3044\u307e\u3059\u3002<\/p>\n<p>\u305d\u3053\u3067\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3057\u3066\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5f15\u304d\u7b97\u3092\u3057\u307e\u3059\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/hatsudy.com\/jp\/wp-content\/uploads\/2023\/01\/nchw4.jpg\" alt=\"\" width=\"500\" height=\"190\" class=\"aligncenter size-full wp-image-12077\" \/><\/p>\n<p>\u306a\u304a\\(\u03b1=3\u03b1+4\\)\u3067\u3042\u308b\u305f\u3081\u3001\\(\u03b1=-2\\)\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5f0f\u3092\u5909\u5f62\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(a_{n+1}-\u03b1=3(a_n-\u03b1)\\)<\/p>\n<p>\\(a_{n+1}+2=3(a_n+2)\\)<\/p>\n<p>\u3053\u3053\u3067\u3001\\(a_n+2=b_n\\)\u3068\u3057\u307e\u3057\u3087\u3046\u3002\u305d\u3046\u3059\u308b\u3068\u3001\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u308f\u308a\u307e\u3059\u3002<\/p>\n<p>\\(b_{n+1}=3b_n\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\\(b_n\\)\u306f\u516c\u6bd43\u306e\u7b49\u6bd4\u6570\u5217\u3068\u308f\u304b\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u3001\\(b_n\\)\u306e\u521d\u9805\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(b_1=a_1+2\\)<\/p>\n<p>\\(b_1=1+2\\)<\/p>\n<p>\\(b_1=3\\)<\/p>\n<p>\\(b_n\\)\u306e\u521d\u9805\u306f3\u3067\u3042\u308b\u305f\u3081\u3001\u4e00\u822c\u9805\u306f\\(b_n=3\u00b73^{n-1}\\)\\(=3^n\\)\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3059\u308b\u3053\u3068\u3067\u4e00\u822c\u9805\\(a_n\\)\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\\(b_n=3^n\\)<\/p>\n<p>\\(a_n+2=3^n\\)<\/p>\n<p>\\(a_n=3^n-2\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3067\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<h3>\u306a\u305c\u7570\u306a\u308b\u6570\u5b57\u3092\u03b1\u306b\u7f6e\u304d\u63db\u3048\u308b\u306e\u304b\uff1f<\/h3>\n<p>\u306a\u304a\u6f38\u5316\u5f0f\u306e\u8a08\u7b97\u3067\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5b66\u3076\u3068\u304d\u3001\u591a\u304f\u306e\u4eba\u304c\u6df7\u4e71\u3057\u307e\u3059\u3002\\(a_{n+1}\\)\u3068\\(a_n\\)\u306f\u7570\u306a\u308b\u5024\u3067\u3059\u3002\u305d\u308c\u306b\u3082\u95a2\u308f\u3089\u305a\u3001\\(a_{n+1}=\u03b1\\)\u3001\\(a_n=\u03b1\\)\u3068\u8003\u3048\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002\u306a\u305c\u3001\u3053\u306e\u3088\u3046\u306a\u8a08\u7b97\u304c\u6210\u308a\u7acb\u3064\u306e\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<p>\u6f38\u5316\u5f0f\u3067\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3059\u308b\u306e\u306f\u3001\u7b49\u6bd4\u6570\u5217\u3078\u5909\u5f62\u3059\u308b\u305f\u3081\u3067\u3059\u3002\u5148\u307b\u3069\u8a18\u3057\u305f\u554f\u984c\u3092\u4ee5\u4e0b\u306b\u518d\u3073\u63b2\u8f09\u3057\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_{n+1}=3a_n+4\\) &#8211; \u2460<\/li>\n<\/ul>\n<p>\u3053\u306e\u5f0f\u3067\u90aa\u9b54\u306a\u6570\u5b57\u304c4\u3067\u3059\u30024\u304c\u306a\u3051\u308c\u3070\u3001\u5f0f\u306f\u7b49\u6bd4\u6570\u5217\u3068\u306a\u308a\u307e\u3059\u3002\u7b49\u6bd4\u6570\u5217\u578b\u3067\u3042\u308c\u3070\u3001\u5bb9\u6613\u306b\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067<span style=\"color: #0000ff;\">\u7b49\u6bd4\u6570\u5217\u306b\u5909\u5f62\u3059\u308b\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u7121\u7406\u3084\u308a\u5f62\u3092\u5909\u3048\u307e\u3057\u3087\u3046\u3002<\/span><\/p>\n<ul>\n<li>\\(a_{n+1}-\u03b1=3(a_n-\u03b1)\\) &#8211; \u2461<\/li>\n<\/ul>\n<p>\u03b1\u306e\u5024\u304c\u4f55\u304b\u306f\u4e0d\u660e\u3067\u3059\u3002\u305f\u3060\u3044\u305a\u308c\u306b\u3057\u3066\u3082\u3001\u5148\u307b\u3069\u89e3\u8aac\u3057\u305f\u901a\u308a\u3001\u3053\u306e\u5f62\u3067\u3042\u308c\u3070\u7b49\u6bd4\u6570\u5217\u578b\u306e\u6f38\u5316\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u4e00\u822c\u5f0f\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\u6b21\u306b\u3001\u2460\u304b\u3089\u2461\u3092\u5f15\u304d\u307e\u3057\u3087\u3046\u3002\u305d\u3046\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(\u03b1=3\u03b1+4\\)<\/li>\n<\/ul>\n<p>\u3053\u3046\u3057\u3066\u3001\u5143\u306e\u5f0f\u306b\u3064\u3044\u3066\\(a_{n+1}\\)\u3068\\(a_n\\)\u3092\u03b1\u3078\u7f6e\u304d\u63db\u3048\u305f\u5834\u5408\u3068\u540c\u3058\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u7279\u6027\u65b9\u7a0b\u5f0f\u3068\u3044\u3046\u306e\u306f\u3001\u7570\u306a\u308b\u6570\u5b57\u3092\u03b1\u306b\u3057\u3066\u3044\u308b\u308f\u3051\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u5f0f\u3092\u7b49\u6bd4\u6570\u5217\u306b\u3059\u308b\u305f\u3081\u3001\u5229\u7528\u3055\u308c\u308b\u5f0f\u304c\u7279\u6027\u65b9\u7a0b\u5f0f\u3067\u3059\u3002<\/p>\n<p>\u305f\u3060\u3001\u30d2\u30f3\u30c8\u306a\u3057\u3067\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5f97\u308b\u306e\u306f\u96e3\u3057\u3044\u3067\u3059\u3002\u3053\u306e\u3068\u304d\u6f38\u5316\u5f0f\u304c\\(a_{n+1}=pa_n+q\\)\u306e\u5f62\u306a\u306e\u3067\u3042\u308c\u3070\u3001\\(a_{n+1}=\u03b1\\)\u3001\\(a_n=\u03b1\\)\u306b\u3059\u308b\u3053\u3068\u3067\u3001\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u4f5c\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u3063\u3066\u3044\u307e\u3059\u3002\u305d\u3053\u3067\u3001\u3053\u3046\u3057\u305f\u7f6e\u304d\u63db\u3048\u3092\u3059\u308b\u306e\u3067\u3059\u3002<\/p>\n<h3>\u5b9a\u6570\u3067\u306f\u306a\u304f\u5f0f\u306b\\(n\\)\u3092\u542b\u3080\u5834\u5408\u3001\u968e\u5dee\u6570\u5217\u3092\u7528\u3044\u3066\u554f\u984c\u3092\u89e3\u304f<\/h3>\n<p>\u306a\u304a\u5834\u5408\u306b\u3088\u3063\u3066\u306f\u3001\u968e\u5dee\u6570\u5217\u3068\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u7528\u3044\u3066\u4e00\u822c\u5f0f\u3092\u5f97\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002\u6f38\u5316\u5f0f\u306e\u4e2d\u306b\u306f\u3001\u7b49\u6bd4\u6570\u5217\u306b\u52a0\u3048\u3066\u300c\\(n\\)\u3092\u542b\u3080\u5f0f\u300d\u304c\u52a0\u308f\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u3002\u3064\u307e\u308a\u3001\u7b49\u6bd4\u6570\u5217\u306b\u5b9a\u6570\u304c\u52a0\u308f\u308b\u306e\u3067\u306f\u306a\u304f\u3001\u7b49\u6bd4\u6570\u5217\u3068\\(n\\)\u3092\u542b\u3080\u5f0f\u3092\u542b\u3093\u3067\u3044\u308b\u306e\u3067\u3059\u3002<\/p>\n<p>\u4f8b\u3048\u3070\u3001\u4ee5\u4e0b\u306e\u554f\u984c\u304c\u8a72\u5f53\u3057\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_1=1\\)\u3001\\(a_{n+1}=3a_n+4n\\)\u306e\u3068\u304d\u3001\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\u3053\u306e\u3088\u3046\u306b\u5b9a\u6570\u3067\u306f\u306a\u304f\u3001\\(n\\)\u3092\u542b\u3080\u5f0f\uff08\u4eca\u56de\u306e\u4f8b\u3067\u306f\\(4n\\)\uff09\u304c\u5b58\u5728\u3057\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u3001\u968e\u5dee\u6570\u5217\u3092\u7528\u3044\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n<p>\\(a_{n+1}=3a_n+4n\\)\u306b\u3064\u3044\u3066\u3001\\(n\\)\u3092\\(n+1\\)\u306b\u5909\u3048\u307e\u3057\u3087\u3046\u3002\u6f38\u5316\u5f0f\u306f\u4ee5\u524d\u306e\u5f0f\u3092\u5229\u7528\u3057\u3066\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u305f\u3081\u3001\\(n\\)\u3092\\(n+1\\)\u306b\u5909\u3048\u308b\u306e\u306f\u554f\u984c\u3042\u308a\u307e\u305b\u3093\u3002\u305d\u3046\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_{n+2}=3a_{n+1}+4(n+1)\\)<\/li>\n<\/ul>\n<p>\u305d\u3053\u3067\\(a_{n+1}=3a_n+4n\\)\u3068\\(a_{n+2}=3a_{n+1}+4(n+1)\\)\u3092\u5229\u7528\u3057\u3066\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5f15\u304d\u7b97\u3092\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/hatsudy.com\/jp\/wp-content\/uploads\/2023\/01\/nchw6.jpg\" alt=\"\" width=\"600\" height=\"150\" class=\"aligncenter size-full wp-image-12080\" \/><\/p>\n<p>\u3053\u306e\u3088\u3046\u306b\u3001\u968e\u5dee\u6570\u5217\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\\(b_n=a_{n+1}-a_n\\)\u3068\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u5f0f\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(b_{n+1}=3b_n+4\\)<\/li>\n<\/ul>\n<p>\u3053\u306e\u5f0f\u3068\u4f3c\u305f\u5f0f\u3092\u5148\u307b\u3069\u8a08\u7b97\u3057\u305f\u3068\u601d\u3044\u307e\u3059\u3002\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u30014\uff08\u5b9a\u6570\uff09\u3092\u6d88\u53bb\u3067\u304d\u307e\u3059\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/hatsudy.com\/jp\/wp-content\/uploads\/2023\/01\/nchw7.jpg\" alt=\"\" width=\"498\" height=\"190\" class=\"aligncenter size-full wp-image-12081\" \/><\/p>\n<p>\u306a\u304a\\(\u03b1=3\u03b1+4\\)\u3067\u3042\u308b\u305f\u3081\u3001\\(\u03b1=-2\\)\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u6570\u5217\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(b_{n+1}+2=3(b_n+2)\\)<\/li>\n<\/ul>\n<p>\u3053\u3046\u3057\u3066\u3001\u6570\u5217\\(\\{b_n+2\\}\\)\u306f\u516c\u6bd43\u306e\u7b49\u6bd4\u6570\u5217\u3068\u308f\u304b\u308a\u307e\u3059\u3002\u307e\u305f\u3001\u521d\u9805\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(b_1=a_{2}-a_1\\)<\/p>\n<p>\\(b_1=3+4-1\\)<\/p>\n<p>\\(b_1=6\\)<\/p>\n<p>\\(b_1=6\\)\u3067\u3042\u308b\u305f\u3081\u3001\u6570\u5217\\(\\{b_n+2\\}\\)\u306e\u521d\u9805\u306f\\(6+2=8\\)\u3067\u3059\u3002\u3064\u307e\u308a\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5f0f\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<p>\\(b_n+2=8\u00b73^{n-1}\\)<\/p>\n<p>\\(b_n=8\u00b73^{n-1}-2\\)<\/p>\n<p>\u6b21\u306b\u3001\u968e\u5dee\u6570\u5217\u306e\u4e00\u822c\u9805\\(b_n\\)\u3092\u5229\u7528\u3057\u3066\u3001\u5143\u306e\u6570\u5217\u306e\u4e00\u822c\u9805\\(a_n\\)\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002\\(a_1=1\\)\u3001\\(b_n=8\u00b73^{n-1}-2\\)\u3067\u3042\u308b\u305f\u3081\u3001\\(n\u22672\\)\u306e\u3068\u304d\u3001\u4ee5\u4e0b\u306e\u8a08\u7b97\u306b\u3088\u3063\u3066\u4e00\u822c\u9805\\(a_n\\)\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\\(a_n=1+\\displaystyle\\sum_{k=1}^{n-1}{(8\u00b73^{k-1}-2)}\\)<\/p>\n<p>\\(a_n=1+\\displaystyle\\frac{8(3^{n-1}-1)}{3-1}\\)\\(-2(n-1)\\)<\/p>\n<p>\\(a_n=4\u00b73^{n-1}-2n-1\\)<\/p>\n<p>\u306a\u304a\\(n=1\\)\u306e\u3068\u304d\u3001\\(a_1=1\\)\u3067\u3059\u3002\\(n=1\\)\u3067\u3042\u3063\u3066\u3082\u6210\u308a\u7acb\u3064\u305f\u3081\u3001\u4e00\u822c\u9805\u306f\\(a_n=4\u00b73^{n-1}-2n-1\\)\u3067\u3059\u3002<\/p>\n<h2>\u7279\u6b8a\u306a\u6f38\u5316\u5f0f\u3068\u30d1\u30bf\u30fc\u30f3<\/h2>\n<p>\u5148\u307b\u3069\u30012\u30d1\u30bf\u30fc\u30f3\u306e\u7279\u6b8a\u306a\u6f38\u5316\u5f0f\u3092\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002\u4ee5\u4e0b\u306e\u6f38\u5316\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u6d3b\u7528\uff1a\\(a_{n+1}=pa_n+q\\)<\/li>\n<li>\u968e\u5dee\u6570\u5217\u3092\u5229\u7528\uff1a\\(a_{n+1}=pa_n+f(n)\\)<\/li>\n<\/ul>\n<p>\u7279\u6b8a\u306a\u6f38\u5316\u5f0f\u306f\u307b\u304b\u306b\u3082\u30d1\u30bf\u30fc\u30f3\u304c\u3042\u308a\u307e\u3059\u3002\u3069\u306e\u6f38\u5316\u5f0f\u306b\u3064\u3044\u3066\u3082\u3001\u89e3\u304d\u65b9\u304c\u6c7a\u307e\u3063\u3066\u304a\u308a\u3001\u89e3\u304d\u65b9\u3092\u77e5\u3063\u3066\u3044\u308c\u3070\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u4ee5\u4e0b\u304c\u7279\u6b8a\u306a\u6f38\u5316\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\u6307\u6570\u306b\\(n\\)\u3092\u542b\u3080\uff1a\\(a_{n+1}=pa_n+q^n\\)<\/li>\n<li>\u6f38\u5316\u5f0f\u304c\u5206\u6570\u3092\u542b\u3080\uff1a\\(a_{n+1}=\\displaystyle\\frac{a_n}{pa_n+q}\\)<\/li>\n<li>\\(a_n\\)\u306b\u30eb\u30fc\u30c8\u307e\u305f\u306f\u7d2f\u4e57\u304c\u542b\u307e\u308c\u308b\uff1a\\(a_{n+1}=pa_n^q\\)<\/li>\n<li>\\(a_n\\)\u306e\u4fc2\u6570\u306b\\(n\\)\u304c\u542b\u307e\u308c\u308b\uff1a\\(a_{n+1}=f(n)\u00b7a_n+q\\)<\/li>\n<li>\\(a_n\\)\u306e\u4fc2\u6570\u306b\\(n\\)\u304c\u542b\u307e\u308c\u308b\uff1a\\(a_{n+1}=f(n)\u00b7a_n\\)<\/li>\n<\/ul>\n<p>\u6f38\u5316\u5f0f\u3092\u5b66\u3076\u3068\u304d\u3001\u3059\u3079\u3066\u306e\u30d1\u30bf\u30fc\u30f3\u3092\u89e3\u3051\u308b\u3088\u3046\u306b\u306a\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u89e3\u304d\u65b9\u3092\u77e5\u3063\u3066\u3044\u308b\u304b\u3069\u3046\u304b\u304c\u91cd\u8981\u306b\u306a\u308b\u305f\u3081\u3001\u305d\u308c\u305e\u308c\u306e\u30d1\u30bf\u30fc\u30f3\u3092\u899a\u3048\u307e\u3057\u3087\u3046\u3002\u524d\u8ff0\u306e\u901a\u308a\u3001\u30d2\u30f3\u30c8\u306a\u3057\u3067\u6f38\u5316\u5f0f\u306e\u554f\u984c\u3092\u89e3\u304f\u306e\u306f\u96e3\u3057\u3044\u304b\u3089\u3067\u3059\u3002<\/p>\n<h3>\u6307\u6570\u306b\\(n\\)\u3092\u542b\u3080\u6f38\u5316\u5f0f\u306f\u4e21\u8fba\u3092\u6307\u6570\u3067\u5272\u308b<\/h3>\n<p>\\(a_{n+1}=pa_n+q^n\\)\u578b\u306e\u6f38\u5316\u5f0f\u306b\u3064\u3044\u3066\u3001\u5f0f\u3092\\(a_{n+1}=pa_n+q\\)\u578b\u306b\u5909\u3048\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3057\u3087\u3046\u3002\u3053\u308c\u3092\u3059\u308b\u305f\u3081\u3001<span style=\"color: #ff0000;\">\u4e21\u8fba\u3092\\(q^{n+1}\\)\u3067\u5272\u308a\u307e\u3059\u3002<\/span>\u3053\u308c\u306b\u3088\u308a\u3001\u6307\u6570\u306b\\(n\\)\u304c\u542b\u307e\u308c\u306a\u304f\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u4ee5\u4e0b\u306e\u554f\u984c\u3092\u89e3\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(a_1=4\\)\u3001\\(a_{n+1}=3a_n+2^{n+1}\\)\u306b\u3088\u3063\u3066\u5b9a\u3081\u3089\u308c\u308b\u6570\u5217\\(\\{a_n\\}\\)\u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\u5f0f\u3092\\(a_{n+1}=pa_n+q\\)\u578b\u306b\u5909\u3048\u308b\u305f\u3081\u3001\u4e21\u8fba\u3092\\(2^{n+1}\\)\u3067\u5272\u308a\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>\\(\\displaystyle\\frac{a_{n+1}}{2^{n+1}}=\\displaystyle\\frac{3}{2}\u00b7\\displaystyle\\frac{a_n}{2^n}+1\\)<\/li>\n<\/ul>\n<p>\u305d\u3053\u3067\u3001\\(b_n=\\displaystyle\\frac{a_n}{2^n}\\)\u3068\u3057\u307e\u3057\u3087\u3046\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u5f0f\u304c\u308f\u304b\u308a\u3084\u3059\u304f\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(b_{n+1}=\\displaystyle\\frac{3}{2}b_n+1\\)<\/li>\n<\/ul>\n<p>\u7279\u6027\u65b9\u7a0b\u5f0f\\(\u03b1=\\displaystyle\\frac{3}{2}\u03b1+1\\)\u306b\u3064\u3044\u3066\u3001\u5f0f\u3092\u89e3\u304f\u3068\\(\u03b1=-2\\)\u3067\u3059\u3002\u307e\u305f\u3001\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3067\u5f0f\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(b_{n+1}+2=\\displaystyle\\frac{3}{2}(b_n+2)\\)<\/li>\n<\/ul>\n<p>\u3053\u3046\u3057\u3066\u3001\u6570\u5217\\(\\{b_n+2\\}\\)\u306f\u516c\u6bd4\\(\\displaystyle\\frac{3}{2}\\)\u306e\u7b49\u6bd4\u6570\u5217\u3068\u308f\u304b\u308a\u307e\u3057\u305f\u3002\u307e\u305f\\(b_1=\\displaystyle\\frac{a_1}{2^1}\\)\\(=2\\)\u3067\u3059\u3002\\(b_1+2=4\\)\u3067\u3042\u308b\u305f\u3081\u3001\u521d\u9805\u306f4\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u4e00\u822c\u9805\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<p>\\(b_n+2=4\u00b7\\left(\\displaystyle\\frac{3}{2}\\right)^{n-1}\\)<\/p>\n<p>\\(b_n=4\u00b7\\left(\\displaystyle\\frac{3}{2}\\right)^{n-1}-2\\)<\/p>\n<p>\\(\\displaystyle\\frac{a_n}{2^n}=4\u00b7\\left(\\displaystyle\\frac{3}{2}\\right)^{n-1}-2\\)<\/p>\n<p>\\(a_n=2^n\u00b74\u00b7\\left(\\displaystyle\\frac{3}{2}\\right)^{n-1}-2\u00b72^n\\)<\/p>\n<p>\\(a_n=8\u00b73^{n-1}-2^{n+1}\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<h3>\u5206\u6570\u3092\u542b\u3080\u6f38\u5316\u5f0f\u3067\u9006\u6570\u3092\u5229\u7528\u3059\u308b<\/h3>\n<p>\\(a_{n+1}=\\displaystyle\\frac{a_n}{pa_n+q}\\)\u306e\u3088\u3046\u306b\u3001\u5206\u5b50\u306b\\(a_n\\)\u306e\u307f\u304c\u542b\u307e\u308c\u3066\u3044\u308b\u5834\u5408\u3001\u9006\u6570\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u5f0f\u3092\\(a_{n+1}=pa_n+q\\)\u578b\u306b\u5909\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u306a\u304a<span style=\"color: #0000ff;\">\u9006\u6570\u3092\u5229\u7528\u3059\u308b\u305f\u3081\u3001\\(a_n\u22600\\)\u3068\u793a\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/span><\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u4ee5\u4e0b\u306e\u554f\u984c\u306e\u7b54\u3048\u306f\u4f55\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<ul>\n<li>\\(a_1=1\\)\u3001\\(a_{n+1}=\\displaystyle\\frac{a_n}{2a_n+1}\\)\u306b\u3088\u3063\u3066\u5b9a\u3081\u3089\u308c\u308b\u6570\u5217\\(\\{a_n\\}\\)\u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\u3053\u306e\u3088\u3046\u306b\u5206\u5b50\u3068\u5206\u6bcd\u306b\\(a_n\\)\u3092\u542b\u3080\u5834\u5408\u3001\u9006\u6570\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3067\u8a08\u7b97\u3067\u304d\u308b\u30b1\u30fc\u30b9\u304c\u3042\u308a\u307e\u3059\u3002\\(a_1=1\\)\u3067\u3042\u308a\u3001\\(\\displaystyle\\frac{a_n}{2a_n+1}\\)\u3067\u306f\u5206\u6bcd\u306b\u5f15\u304d\u7b97\u304c\u3042\u308a\u307e\u305b\u3093\u3002\u305d\u306e\u305f\u3081\\(a_n\\)\u306f\u5e38\u306b\u6b63\u306e\u5024\u3067\u3042\u308a\u3001\\(a_n\u22600\\)\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u9006\u6570\u3092\u5229\u7528\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(\\displaystyle\\frac{1}{a_{n+1}}=\\displaystyle\\frac{1}{a_n}+2\\)<\/li>\n<\/ul>\n<p>\u307e\u305f\u3001\\(\\displaystyle\\frac{1}{a_n}=b_n\\)\u3068\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(b_{n+1}=b_n+2\\)<\/p>\n<p>\\(b_{n+1}-b_n=2\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u6570\u5217\\(\\{b_n\\}\\)\u306f\u516c\u5dee2\u306e\u7b49\u5dee\u6570\u5217\u3068\u308f\u304b\u308a\u307e\u3059\u3002\u307e\u305f\u521d\u9805\u306f\\(b_1=\\displaystyle\\frac{1}{a_1}\\)\\(=1\\)\u3067\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>\\(b_n=1+(n-1)2\\)<\/p>\n<p>\\(b_n=2n-1\\)<\/p>\n<p>\u306a\u304a\\(\\displaystyle\\frac{1}{a_n}=b_n\\)\u3067\u3042\u308b\u305f\u3081\u3001\\(\\displaystyle\\frac{1}{a_n}=2n-1\\)\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\\(a_n=\\displaystyle\\frac{1}{2n-1}\\)\u304c\u7b54\u3048\u3068\u308f\u304b\u308a\u307e\u3059\u3002\u5206\u6570\u3092\u542b\u3080\u6f38\u5316\u5f0f\u3067\u306f\u3001\u9006\u6570\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3067\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u304b\u3069\u3046\u304b\u78ba\u304b\u3081\u307e\u3057\u3087\u3046\u3002<\/p>\n<h3>\\(a_n\\)\u306b\u30eb\u30fc\u30c8\u307e\u305f\u306f\u7d2f\u4e57\u304c\u542b\u307e\u308c\u308b\u5834\u5408\u3001\u5bfe\u6570\u3092\u5229\u7528\u3059\u308b<\/h3>\n<p>\\(\\sqrt{a_n}\\)\u306e\u3088\u3046\u306b\u3001\\(a_n\\)\u304c\u30eb\u30fc\u30c8\u3092\u542b\u3080\u30b1\u30fc\u30b9\u304c\u3042\u308a\u307e\u3059\u3002\u307e\u305f\u306f\u3001\\(a_n^2\\)\u3084\\(a_n^3\\)\u306e\u3088\u3046\u306b\u3001\\(a_n\\)\u306b\u7d2f\u4e57\u304c\u5b58\u5728\u3059\u308b\u30b1\u30fc\u30b9\u3082\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u3001\u5bfe\u6570\u3092\u5229\u7528\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p><span style=\"color: #0000ff;\">\u5bfe\u6570\u3092\u52a0\u3048\u308b\u3068\u3001\u7d2f\u4e57\u3092\u6d88\u53bb\u3067\u304d\u307e\u3059\u3002<\/span>\\(\\sqrt{a_n}=a_n^{\\frac{1}{2}}\\)\u3067\u3042\u308b\u305f\u3081\u3001\\(a_n\\)\u304c\u30eb\u30fc\u30c8\u3092\u542b\u3080\u5834\u5408\u3067\u3042\u3063\u3066\u3082\u5bfe\u6570\u306e\u5229\u7528\u306f\u6709\u52b9\u3067\u3059\u3002\u5bfe\u6570\u3092\u5229\u7528\u3059\u308c\u3070\u3001\\(a_{n+1}=pa_n+q\\)\u578b\u3078\u5909\u5f62\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\u5bfe\u6570\u3092\u5229\u7528\u3059\u308b\u305f\u3081\u3001\\(a_n\u22600\\)\u3067\u3042\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u307e\u305f\u771f\u6570\u6761\u4ef6\u3082\u78ba\u8a8d\u3057\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\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>\\(a_1=1\\)\u3001\\(a_{n+1}=2\\sqrt{a_n}\\)\u306b\u3088\u3063\u3066\u5b9a\u3081\u3089\u308c\u308b\u6570\u5217\\(\\{a_n\\}\\)\u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\\(a_1=1\\)\u3067\u3042\u308b\u305f\u3081\u3001\\(a_{n+1}=2\\sqrt{a_n}\\)\u306f\u5e38\u306b\u6b63\u306e\u5024\u3067\u3059\u3002\\(a_n&gt;0\\)\u3067\u3042\u308a\u3001\u5e95\u30922\u3068\u3059\u308b\u5bfe\u6570\u3092\u4e21\u8fba\u306b\u53d6\u308a\u307e\u3057\u3087\u3046\u3002\u5bfe\u6570\u306e\u5e95\u306b\u3064\u3044\u3066\u306f\u3001\u8a08\u7b97\u3057\u3084\u3059\u3044\u5e95\u3092\u9078\u3073\u307e\u3057\u3087\u3046\u3002\u4eca\u56de\u306e\u4f8b\u984c\u3067\u306f\u4ee5\u4e0b\u306e\u8a08\u7b97\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(log_2a_{n+1}=log_22\\sqrt{a_n}\\)<\/p>\n<p>\\(log_2a_{n+1}=log_22+log_2a_n^{\\frac{1}{2}}\\)<\/p>\n<p>\\(log_2a_{n+1}=\\displaystyle\\frac{1}{2}log_2a_n+1\\)<\/p>\n<p>\\(log_2a_n=b_n\\)\u3068\u3057\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>\\(b_{n+1}=\\displaystyle\\frac{1}{2}b_n+1\\)<\/li>\n<\/ul>\n<p>\u3053\u3046\u3057\u3066\u3001\\(a_{n+1}=pa_n+q\\)\u578b\u3078\u5909\u5f62\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u3053\u306e\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u308c\u3070\u3001\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u4e00\u822c\u9805\u3092\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\\(\u03b1=\\displaystyle\\frac{1}{2}\u03b1+1\\)\u3092\u5229\u7528\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(b_{n+1}-\u03b1=\\displaystyle\\frac{1}{2}(b_n-\u03b1)\\)<\/li>\n<\/ul>\n<p>\u306a\u304a\\(\u03b1=\\displaystyle\\frac{1}{2}\u03b1+1\\)\u3088\u308a\u3001\\(\u03b1=2\\)\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(b_{n+1}-2=\\displaystyle\\frac{1}{2}(b_n-2)\\)<\/li>\n<\/ul>\n<p>\u3053\u3046\u3057\u3066\u3001\u6570\u5217\\(\\{b_n-2\\}\\)\u306f\u516c\u6bd4\\(\\displaystyle\\frac{1}{2}\\)\u306e\u7b49\u6bd4\u6570\u5217\u3068\u308f\u304b\u308a\u307e\u3059\u3002\u307e\u305f\\(b_1=log_2a_1\\)\\(=log_21\\)\\(=0\\)\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u521d\u9805\u306f\\(0-2=-2\\)\u3067\u3059\u3002\u3064\u307e\u308a\u3001\u6570\u5217\\(\\{b_n-2\\}\\)\u306e\u4e00\u822c\u9805\u306f\u4ee5\u4e0b\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(b_n-2=-2\u00b7\\left(\\displaystyle\\frac{1}{2}\\right)^{n-1}\\)<\/p>\n<p>\\(b_n=2-2\u00b7\\left(\\displaystyle\\frac{1}{2}\\right)^{n-1}\\)<\/p>\n<p>\\(log_2a_n=b_n\\)\u3067\u3042\u308b\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(log_2a_n=2-2\u00b7\\left(\\displaystyle\\frac{1}{2}\\right)^{n-1}\\)<\/p>\n<p>\\(log_2a_n=log_22^{2-2\u00b7\\left(\\frac{1}{2}\\right)^{n-1}}\\)<\/p>\n<p>\\(a_n=2^{2-2\u00b7\\left(\\frac{1}{2}\\right)^{n-1}}\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u4e00\u822c\u9805\\(a_n\\)\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<h3>\\(a_n\\)\u306e\u4fc2\u6570\u306b\\(n\\)\u304c\u542b\u307e\u308c\u308b\uff1a\\(a_{n+1}=f(n)\u00b7a_n+q\\)<\/h3>\n<p>\\(a_n\\)\u306e\u4fc2\u6570\u306b\\(n\\)\u304c\u5b58\u5728\u3059\u308b\u5834\u5408\u3001\u7f6e\u304d\u63db\u3048\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3067\\(a_{n+1}=pa_n+q\\)\u578b\u307e\u305f\u306f\\(a_{n+1}=pa_n+f(n)\\)\u578b\u306b\u5909\u5f62\u3067\u304d\u306a\u3044\u304b\u8003\u3048\u307e\u3057\u3087\u3046\u3002\u30d2\u30f3\u30c8\u306a\u3057\u3067\u5f0f\u3092\u5909\u5f62\u3059\u308b\u306e\u306f\u96e3\u3057\u3044\u3067\u3059\u304c\u3001\u5f0f\u3092\u5909\u5f62\u3067\u304d\u308c\u3070\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u4ee5\u4e0b\u306e\u554f\u984c\u306e\u7b54\u3048\u306f\u4f55\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<ul>\n<li>\\(a_1=1\\)\u3001\\(na_{n+1}=2(n+1)a_n\\)\u306b\u3088\u3063\u3066\u5b9a\u3081\u3089\u308c\u308b\u6570\u5217\\(\\{a_n\\}\\)\u306b\u3064\u3044\u3066\u3001\\(b_n=\\displaystyle\\frac{a_n}{n}\\)\u3068\u3059\u308b\u3068\u304d\u3001\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\u524d\u8ff0\u306e\u901a\u308a\u3001\u30d2\u30f3\u30c8\u306a\u3057\u3067\u5f0f\u3092\u5909\u5f62\u3059\u308b\u306e\u306f\u96e3\u3057\u3044\u3082\u306e\u306e\u3001\u554f\u984c\u6587\u306b\u30d2\u30f3\u30c8\u3068\u3057\u3066\\(b_n=\\displaystyle\\frac{a_n}{n}\\)\u304c\u8a18\u3055\u308c\u3066\u3044\u308b\u305f\u3081\u3001\u3053\u306e\u30d2\u30f3\u30c8\u3092\u5229\u7528\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\((n+1)a_n\\)\u306b\u3064\u3044\u3066\u3001\u3069\u306e\u3088\u3046\u306a\u5024\u3067\u5272\u308c\u3070\\(\\displaystyle\\frac{a_n}{n}\\)\u3068\u306a\u308b\u3067\u3057\u3087\u3046\u304b\u3002\\(n(n+1)\\)\u3067\u5272\u308c\u3070\u3001\\((n+1)a_n\\)\u3092\\(\\displaystyle\\frac{a_n}{n}\\)\u3078\u5909\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067\u3001\u5f0f\u306e\u4e21\u8fba\u3092\\(n(n+1)\\)\u3067\u5272\u308a\u307e\u3057\u3087\u3046\u3002\u305d\u3046\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n<p>\\(\\displaystyle\\frac{a_{n+1}}{n+1}=2\\displaystyle\\frac{a_n}{n}\\)<\/p>\n<p>\\(b_{n+1}=2b_n\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u516c\u6bd42\u306e\u7b49\u6bd4\u6570\u5217\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u307e\u305f\\(b_1=\\displaystyle\\frac{a_1}{1}\\)\\(=1\\)\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\\(b_n=2^{n-1}\\)\u3068\u306a\u308a\u307e\u3059\u3002\u306a\u304a\\(b_n=\\displaystyle\\frac{a_n}{n}\\)\u3067\u3042\u308b\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(b_n=2^{n-1}\\)<\/p>\n<p>\\(\\displaystyle\\frac{a_n}{n}=2^{n-1}\\)<\/p>\n<p>\\(a_n=n\u00b72^{n-1}\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u7f6e\u304d\u63db\u3048\u304c\u53ef\u80fd\u306a\u3088\u3046\u306b\\(a_n\\)\u306e\u4fc2\u6570\u3092\u5909\u5f62\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u4e00\u822c\u9805\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<h3>\u6f38\u5316\u5f0f\u306e\u7e70\u308a\u8fd4\u3057\u306b\u3088\u308a\u3001\u4e00\u822c\u9805\u3092\u5f97\u308b<\/h3>\n<p>\\(a_{n+1}=f(n)\u00b7a_n\\)\u578b\u306e\u6f38\u5316\u5f0f\u306f\u5148\u307b\u3069\u89e3\u8aac\u3057\u305f\u5f0f\u3068\u540c\u3058\u3088\u3046\u306b\u3001\\(a_n\\)\u306b\u4fc2\u6570\\(n\\)\u3092\u542b\u307f\u307e\u3059\u3002\u305f\u3060\u3001\\(a_n\\)\u3068\\(a_{n+1}\\)\u4ee5\u5916\u306e\u9805\u304c\u3042\u308a\u307e\u305b\u3093\u3002\u3053\u306e\u5834\u5408\u3001<span style=\"color: #0000ff;\">\u7b49\u5dee\u6570\u5217\u3084\u7b49\u6bd4\u6570\u5217\u3001\u7279\u6027\u65b9\u7a0b\u5f0f\u306a\u3069\u3092\u5229\u7528\u305b\u305a\u3001\u6f38\u5316\u5f0f\u306e\u7e70\u308a\u8fd4\u3057\u306b\u3088\u3063\u3066\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/span><\/p>\n<p>\u5b9f\u969b\u306b\u554f\u984c\u3092\u89e3\u304b\u306a\u3044\u3068\u610f\u5473\u3092\u7406\u89e3\u3067\u304d\u306a\u3044\u3068\u601d\u3046\u306e\u3067\u3001\u4ee5\u4e0b\u306e\u7df4\u7fd2\u554f\u984c\u3092\u89e3\u3044\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(a_1=1\\)\u3001\\((n+1)a_n=na_{n-1}\\)\uff08\\(n\u22672\\)\uff09\u306b\u3088\u3063\u3066\u5b9a\u3081\u3089\u308c\u308b\u6570\u5217\\(\\{a_n\\}\\)\u306b\u3064\u3044\u3066\u3001\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\u307e\u305a\u3001\u4e21\u8fba\u3092\\(n+1\\)\u3067\u5272\u308a\u307e\u3057\u3087\u3046\u3002\u3053\u306e\u5834\u5408\u3001\u5f0f\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_n=\\displaystyle\\frac{n}{n+1}a_{n-1}\\)<\/li>\n<\/ul>\n<p>\u305d\u308c\u3067\u306f\u3001\u3053\u306e\u6f38\u5316\u5f0f\u3092\u5229\u7528\u3057\u3066\\(a_{n-2}\\)\u3092\u542b\u3080\u5f0f\u3092\u4f5c\u308b\u306b\u306f\u3069\u3046\u3059\u308c\u3070\u3044\u3044\u3067\u3057\u3087\u3046\u304b\u3002\u5148\u307b\u3069\u306e\u6f38\u5316\u5f0f\u3092\u5229\u7528\u3059\u308b\u3068\u3001\\(n\\)\u3092\\(n-1\\)\u306b\u5909\u63db\u3059\u308c\u3070\\(a_{n-1}=\\displaystyle\\frac{n-1}{n}a_{n-2}\\)\u3068\u306a\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u3001\\(a_{n-1}=\\displaystyle\\frac{n-1}{n}a_{n-2}\\)\u3092\u4ee3\u5165\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u5f0f\u3092\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_n=\\displaystyle\\frac{n}{n+1}\\)\\(\u00b7\\displaystyle\\frac{n-1}{n}a_{n-2}\\)<\/li>\n<\/ul>\n<p>\u6b21\u306b\u3001\\(a_{n-3}\\)\u3092\u542b\u3080\u5f0f\u3092\u52a0\u3048\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_n=\\displaystyle\\frac{n}{n+1}\\)\\(\u00b7\\displaystyle\\frac{n-1}{n}\\)\\(\u00b7\\displaystyle\\frac{n-2}{n-1}a_{n-3}\\)<\/li>\n<\/ul>\n<p>\u305d\u3053\u3067\u3001\\(a_1\\)\u306b\u306a\u308b\u307e\u3067\u6f38\u5316\u5f0f\u3092\u7e70\u308a\u8fd4\u3057\u307e\u3057\u3087\u3046\u3002\u306a\u304a\u554f\u984c\u6587\u3088\u308a\u3001\\(n\u22672\\)\u3067\u3042\u308b\u305f\u3081\\(n\\)\u306e\u6700\u5c0f\u5024\u306f2\u3067\u3059\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/hatsudy.com\/jp\/wp-content\/uploads\/2023\/01\/ejdh4.jpg\" alt=\"\" width=\"600\" height=\"183\" class=\"aligncenter size-full wp-image-12095\" \/><\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u4e00\u822c\u7684\u306a\u6f38\u5316\u5f0f\u306e\u8a08\u7b97\u3068\u306f\u307e\u3063\u305f\u304f\u7570\u306a\u308b\u65b9\u6cd5\u306b\u3088\u3063\u3066\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u3053\u3046\u3057\u305f\u65b9\u6cd5\u306b\u3088\u3063\u3066\u6f38\u5316\u5f0f\u3092\u89e3\u304f\u30b1\u30fc\u30b9\u304c\u3042\u308b\u3053\u3068\u3092\u5b66\u3073\u307e\u3057\u3087\u3046\u3002<\/p>\n<h2>\u6f38\u5316\u5f0f\u306e\u89e3\u304d\u65b9\u306b\u306f\u8907\u6570\u306e\u7a2e\u985e\u304c\u3042\u308b<\/h2>\n<p>\u6f38\u5316\u5f0f\u306e\u554f\u984c\u3092\u89e3\u304f\u3068\u304d\u3001\u30d1\u30bf\u30fc\u30f3\u304c\u6c7a\u307e\u3063\u3066\u3044\u307e\u3059\u3002\u305d\u308c\u305e\u308c\u306e\u30d1\u30bf\u30fc\u30f3\u306b\u3064\u3044\u3066\u3001\u968e\u5dee\u6570\u5217\u3084\u7d2f\u4e57\u306e\u5272\u308a\u7b97\u3001\u9006\u6570\u3001\u5bfe\u6570\u306a\u3069\u3001\u3055\u307e\u3056\u307e\u306a\u65b9\u6cd5\u306b\u3088\u3063\u3066\u5f0f\u3092\u5909\u5f62\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u306a\u304a\u3001\u5f0f\u3092\\(a_{n+1}=pa_n+q\\)\u578b\u3078\u5909\u5f62\u3067\u304d\u308c\u3070\u3001\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u6f38\u5316\u5f0f\u306e\u8a08\u7b97\u3067\u7279\u6027\u65b9\u7a0b\u5f0f\u306f\u3072\u3093\u3071\u3093\u306b\u5229\u7528\u3055\u308c\u308b\u305f\u3081\u3001\u3069\u306e\u3088\u3046\u306a\u6027\u8cea\u3092\u3082\u3064\u5f0f\u306a\u306e\u304b\u7406\u89e3\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u6f38\u5316\u5f0f\u306e\u8a08\u7b97\u3067\u5fc5\u305a\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3059\u308b\u308f\u3051\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u7279\u6027\u65b9\u7a0b\u5f0f\u306a\u3057\u306b\u6f38\u5316\u5f0f\u3092\u7b49\u5dee\u6570\u5217\u578b\u3084\u7b49\u6bd4\u6570\u5217\u578b\u3078\u5909\u5f62\u3057\u3001\u554f\u984c\u3092\u89e3\u3051\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002\u6f38\u5316\u5f0f\u306e\u7e70\u308a\u8fd4\u3057\u306b\u3088\u308a\u3001\u4e00\u822c\u9805\u3092\u5f97\u3089\u308c\u308b\u30b1\u30fc\u30b9\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n<p>\u30d2\u30f3\u30c8\u306a\u3057\u3067\u6f38\u5316\u5f0f\u306e\u554f\u984c\u3092\u89e3\u304f\u306e\u306f\u96e3\u3057\u3044\u3067\u3059\u3002\u305d\u3053\u3067\u5f0f\u3092\u78ba\u8a8d\u3059\u308b\u3053\u3068\u3067\u3001\u554f\u984c\u3092\u89e3\u304f\u305f\u3081\u306e\u65b9\u91dd\u3092\u7acb\u3066\u3089\u308c\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u3087\u3046\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6f38\u5316\u5f0f\u306b\u95a2\u3059\u308b\u554f\u984c\u3092\u89e3\u304f\u3068\u304d\u3001\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u304c\u591a\u3044\u3067\u3059\u3002\u03b1\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u6f38\u5316\u5f0f\u3092\u5909\u5f62\u3057\u3066\u6570\u5217\u306e\u4e00\u822c\u5f0f\u3092\u5f97\u308b\u306e\u3067\u3059\u3002 \u305f\u3060\u3001\u591a\u304f\u306e\u4eba\u306f\u7279\u6027\u65b9\u7a0b\u5f0f\u3092\u5b66\u3076\u3068\u304d\u306b\u6df7\u4e71\u3057\u307e\u3059\u3002\u7570\u306a\u308b\u6570\u5b57\u3092\u03b1\u306b\u7f6e\u304d\u63db\u3048\u308b\u305f\u3081\u3001\u306a\u305c [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":12096,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":{"0":"post-12075","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\/12075","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=12075"}],"version-history":[{"count":12,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/posts\/12075\/revisions"}],"predecessor-version":[{"id":12193,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/posts\/12075\/revisions\/12193"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/media\/12096"}],"wp:attachment":[{"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/media?parent=12075"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/categories?post=12075"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/tags?post=12075"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}