{"id":12115,"date":"2023-01-07T15:06:59","date_gmt":"2023-01-07T06:06:59","guid":{"rendered":"https:\/\/hatsudy.com\/jp\/?p=12115"},"modified":"2023-01-14T09:17:58","modified_gmt":"2023-01-14T00:17:58","slug":"system-of-recurrence-relations","status":"publish","type":"post","link":"https:\/\/hatsudy.com\/jp\/system-of-recurrence-relations.html","title":{"rendered":"\u9023\u7acb\u6f38\u5316\u5f0f\uff1a\u4e00\u822c\u9805\u3092\u5f97\u308b\u89e3\u304d\u65b9\u3084\u516c\u5f0f\u306e\u5229\u7528\u6cd5"},"content":{"rendered":"\n<p>\u9023\u7acb\u6f38\u5316\u5f0f\u3067\u306f\u3001\u7570\u306a\u308b\u6f38\u5316\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u9023\u7acb\u6f38\u5316\u5f0f\u3067\u306f\u89e3\u304d\u65b9\u304c\u6c7a\u307e\u3063\u3066\u3044\u308b\u305f\u3081\u3001\u7b54\u3048\u3092\u5f97\u308b\u65b9\u6cd5\u3092\u899a\u3048\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002<\/p>\n<p>\u306a\u304a\u3001\u9023\u7acb\u6f38\u5316\u5f0f\u306b\u306f3\u3064\u306e\u89e3\u304d\u65b9\u304c\u3042\u308a\u307e\u3059\u3002\u3069\u306e\u65b9\u6cd5\u3092\u5229\u7528\u3057\u3066\u3082\u554f\u984c\u3042\u308a\u307e\u305b\u3093\u3002\u307e\u305f\u3001\u7570\u306a\u308b\u65b9\u6cd5\u3092\u5229\u7528\u3057\u3066\u540c\u3058\u89e3\u3092\u5f97\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u7b54\u3048\u306e\u898b\u76f4\u3057\u3068\u3057\u3066\u5229\u7528\u3057\u3066\u3082\u3044\u3044\u3067\u3059\u3002<\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u9023\u7acb\u6f38\u5316\u5f0f\u306f\u3069\u306e\u3088\u3046\u306a\u5f0f\u306a\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u307e\u305f\u3001\u3069\u306e\u3088\u3046\u306b\u4e00\u822c\u9805\u3092\u8a08\u7b97\u3059\u308c\u3070\u3044\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u9023\u7acb\u6f38\u5316\u5f0f\u306e\u554f\u984c\u3092\u89e3\u304f\u65b9\u6cd5\u3092\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n<h2>\u7570\u306a\u308b\u6f38\u5316\u5f0f\u3092\u5229\u7528\u3059\u308b\u9023\u7acb\u6f38\u5316\u5f0f\uff1a\u5bfe\u79f0\u578b\u306e\u8a08\u7b97<\/h2>\n<p>2\u3064\u306e\u5f0f\u3092\u5229\u7528\u3057\u3066\u7b54\u3048\u3092\u5f97\u308b\u65b9\u6cd5\u306b\u9023\u7acb\u65b9\u7a0b\u5f0f\u304c\u3042\u308a\u307e\u3059\u3002\u9023\u7acb\u6f38\u5316\u5f0f\u3082\u540c\u69d8\u3067\u3042\u308a\u30012\u3064\u306e\u6f38\u5316\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<\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u3069\u306e\u3088\u3046\u306b\u8003\u3048\u3066\u9023\u7acb\u6f38\u5316\u5f0f\u3092\u89e3\u3051\u3070\u3044\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u6700\u3082\u5358\u7d14\u306a\u9023\u7acb\u6f38\u5316\u5f0f\u3068\u3057\u3066\u306f\u3001\u5bfe\u79f0\u578b\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f\u304c\u3042\u308a\u307e\u3059\u3002\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3001\u5f0f\u304c\u5bfe\u79f0\u306b\u306a\u3063\u3066\u3044\u308b\u5834\u5408\u304c\u8a72\u5f53\u3057\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_{n+1}=\\color{red}{p}a_n+\\color{red}{q}b_n\\)<\/li>\n<li>\\(b_{n+1}=\\color{red}{q}a_n+\\color{red}{p}b_n\\)<\/li>\n<\/ul>\n<p>\u3053\u306e\u5834\u5408\u3001\u8db3\u3057\u7b97\u3068\u5f15\u304d\u7b97\u306e\u4e21\u65b9\u3092\u3059\u308b\u3053\u3068\u30672\u3064\u306e\u6570\u5217\u3092\u4f5c\u308a\u3001\u4e00\u822c\u9805\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002\u4f8b\u984c\u3068\u3057\u3066\u3001\u4ee5\u4e0b\u306e\u554f\u984c\u3092\u89e3\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(a_1=4\\)\u3001\\(b_1=1\\)\u3001\\(a_{n+1}=3a_n+b_n\\)\u3001\\(b_{n+1}=a_n+3b_n\\)\u306b\u3088\u3063\u3066\u5b9a\u3081\u3089\u308c\u308b\u6570\u5217\\(\\{a_n\\}\\)\u3068\\(\\{b_n\\}\\)\u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\u4ee5\u4e0b\u306e2\u3064\u306e\u6f38\u5316\u5f0f\u3092\u5229\u7528\u3057\u3066\u8db3\u3057\u7b97\u3068\u5f15\u304d\u7b97\u3092\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(a_{n+1}=3a_n+b_n\\) &#8211; \u2460<\/li>\n<li>\\(b_{n+1}=a_n+3b_n\\) &#8211; \u2461<\/li>\n<\/ul>\n<p>\u305d\u306e\u5f8c\u3001\u305d\u308c\u305e\u308c\u306e\u6570\u5217\u306b\u3064\u3044\u3066\u4e00\u822c\u9805\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n<p><strong>\u30fb\\(\u2460+\u2461\\)\u3088\u308a<\/strong><\/p>\n<ul>\n<li>\\(a_{n+1}+b_{n+1}=4(a_n+b_n)\\)<\/li>\n<\/ul>\n<p>\u6570\u5217\\(\\{a_n+b_n\\}\\)\u306f\u521d\u9805\\(a_1+b_1=5\\)\u3001\u516c\u6bd44\u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\u3001\u4e00\u822c\u9805\u306f\u4ee5\u4e0b\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_n+b_n=5\u00b74^{n-1}\\) &#8211; \u2462<\/li>\n<\/ul>\n<p><strong>\u30fb\\(\u2460-\u2461\\)\u3088\u308a<\/strong><\/p>\n<ul>\n<li>\\(a_{n+1}-b_{n+1}=2(a_n-b_n)\\)<\/li>\n<\/ul>\n<p>\u6570\u5217\\(\\{a_n-b_n\\}\\)\u306f\u521d\u9805\\(a_1-b_1=3\\)\u3001\u516c\u6bd42\u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\u3001\u4e00\u822c\u9805\u306f\u4ee5\u4e0b\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_n-b_n=3\u00b72^{n-1}\\) &#8211; \u2463<\/li>\n<\/ul>\n<p>\u6b21\u306b\u3001\u2462\u3068\u2463\u3092\u5229\u7528\u3059\u308b\u3068\u6570\u5217\\(\\{a_n\\}\\)\u3068\\(\\{b_n\\}\\)\u306e\u4e00\u822c\u9805\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p><strong>\u30fb\\(\u2462+\u2463\\)\u3088\u308a<\/strong><\/p>\n<p>\\(2a_n=5\u00b74^{n-1}+3\u00b72^{n-1}\\)<\/p>\n<p>\\(a_n=\\displaystyle\\frac{5}{2}\u00b74^{n-1}+\\displaystyle\\frac{3}{2}\u00b72^{n-1}\\)<\/p>\n<p><strong>\u30fb\\(\u2462-\u2463\\)\u3088\u308a<\/strong><\/p>\n<p>\\(2b_n=5\u00b74^{n-1}-3\u00b72^{n-1}\\)<\/p>\n<p>\\(b_n=\\displaystyle\\frac{5}{2}\u00b74^{n-1}-\\displaystyle\\frac{3}{2}\u00b72^{n-1}\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\\(a_n\\)\u3068\\(b_n\\)\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u30022\u3064\u306e\u6f38\u5316\u5f0f\u304c\u5bfe\u79f0\u306b\u306a\u3063\u3066\u3044\u308b\u5834\u5408\u3001\u8a08\u7b97\u306f\u96e3\u3057\u304f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n<h2>\u9023\u7acb\u6f38\u5316\u5f0f\u3067\u91cd\u8981\u306a2\u901a\u308a\u306e\u89e3\u304d\u65b9<\/h2>\n<p>\u305f\u30602\u3064\u306e\u5f0f\u304c\u5bfe\u79f0\u3067\u306f\u306a\u3044\u30b1\u30fc\u30b9\u304c\u3072\u3093\u3071\u3093\u306b\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u3001\u3069\u306e\u3088\u3046\u306b\u9023\u7acb\u6f38\u5316\u5f0f\u306e\u7b54\u3048\u3092\u5f97\u308c\u3070\u3044\u3044\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u7b54\u3048\u3092\u5f97\u308b\u305f\u3081\u306b\u306f\u3001\u4ee5\u4e0b\u306e2\u901a\u308a\u306e\u65b9\u6cd5\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n<ul>\n<li>2\u3064\u306e\u6f38\u5316\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u7b49\u6bd4\u6570\u5217\u578b\u3078\u5909\u5f62\u3059\u308b<\/li>\n<li>\u96a3\u63a53\u9805\u9593\u306e\u6f38\u5316\u5f0f\u3078\u5909\u5f62\u3059\u308b<\/li>\n<\/ul>\n<p>\u3069\u3061\u3089\u306e\u65b9\u6cd5\u3092\u5229\u7528\u3057\u3066\u3082\u554f\u984c\u3042\u308a\u307e\u305b\u3093\u304c\u3001\u4e21\u65b9\u3092\u5229\u7528\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u3087\u3046\u3002<\/p>\n<h3>\u6f38\u5316\u5f0f\u3092\u7b49\u6bd4\u6570\u5217\u578b\u3078\u5909\u5f62\u3059\u308b<\/h3>\n<p>\u7b49\u6bd4\u6570\u5217\u578b\u3078\u5909\u5f62\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u6570\u5217\u306e\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067\u30012\u3064\u306e\u6f38\u5316\u5f0f\u3092\u4ee5\u4e0b\u306e\u5f62\u3078\u5909\u5f62\u3057\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_{n+1}+\u03b1b_{n+1}=\u03b2(a_n+\u03b1b_n)\\)<\/li>\n<\/ul>\n<p>\u3053\u306e\u5f62\u3067\u3042\u308c\u3070\u3001\u5148\u307b\u3069\u8a08\u7b97\u3057\u305f\u7b49\u6bd4\u6570\u5217\u578b\u306e\u6f38\u5316\u5f0f\u3068\u540c\u3058\u3068\u308f\u304b\u308a\u307e\u3059\u3002\u306a\u304a\u3001\u3053\u308c\u3092\u516c\u5f0f\u3068\u3057\u3066\u899a\u3048\u308b\u5fc5\u8981\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u7b49\u6bd4\u6570\u5217\u578b\u306e\u6f38\u5316\u5f0f\u3092\u4f5c\u308c\u3070\u3044\u3044\u3068\u7406\u89e3\u3059\u308c\u3070\u3001\u5fc5\u305a\u3053\u306e\u5f0f\u306e\u5f62\u306b\u3057\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u3068\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p>\u305d\u308c\u3067\u306f\u3001\u4ee5\u4e0b\u306e\u7df4\u7fd2\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u306b\u3088\u308a\u3001\u8a08\u7b97\u65b9\u6cd5\u3092\u5b66\u3073\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(a_1=1\\)\u3001\\(b_1=2\\)\u3001\\(a_{n+1}=2a_n+b_n\\)\u3001\\(b_{n+1}=4a_n-b_n\\)\u306b\u3088\u3063\u3066\u5b9a\u3081\u3089\u308c\u308b\u6570\u5217\\(\\{a_n\\}\\)\u3068\\(\\{b_n\\}\\)\u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\u307e\u305a\u3001\\(a_{n+1}+\u03b1b_{n+1}=\u03b2(a_n+\u03b1b_n)\\)\u306e\u5de6\u8fba\u306b\u7740\u76ee\u3057\u307e\u3057\u3087\u3046\u3002\u5de6\u8fba\u306b\\(a_{n+1}=2a_n+b_n\\)\u3068\\(b_{n+1}=4a_n-b_n\\)\u3092\u4ee3\u5165\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(a_{n+1}+\u03b1b_{n+1}\\)<\/p>\n<p>\\(=2a_n+b_n+\u03b1(4a_n-b_n)\\)<\/p>\n<p>\\(=(2+4\u03b1)a_n+(1-\u03b1)b_n\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u5f0f\u3068\u3057\u3066\\(a_{n+1}+\u03b1b_{n+1}=(2+4\u03b1)a_n+(1-\u03b1)b_n\\)\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n<p>\u307e\u305f\\(a_{n+1}+\u03b1b_{n+1}=\u03b2(a_n+\u03b1b_n)\\)\u3067\u3042\u308b\u305f\u3081\u3001\\(a_{n+1}+\u03b1b_{n+1}=\u03b2a_n+\u03b1\u03b2b_n\\)\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u6761\u4ef6\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\((2+4\u03b1)a_n+(1-\u03b1)b_n=\u03b2a_n+\u03b1\u03b2b_n\\)<\/li>\n<\/ul>\n<p>\u3053\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(2+4\u03b1=\u03b2\\)<\/li>\n<li>\\(1-\u03b1=\u03b1\u03b2\\)<\/li>\n<\/ul>\n<p>\u305d\u3053\u3067\u3001\u03b1\u3068\u03b2\u306e\u5024\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002\\(2+4\u03b1=\u03b2\\)\u3092\\(1-\u03b1=\u03b1\u03b2\\)\u3078\u4ee3\u5165\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(1-\u03b1=\u03b1\u03b2\\)<\/p>\n<p>\\(1-\u03b1=\u03b1(2+4\u03b1)\\)<\/p>\n<p>\\(4\u03b1^2+3\u03b1-1=0\\)<\/p>\n<p>\\((4\u03b1-1)(\u03b1+1)=0\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\\(\u03b1=-1,\\displaystyle\\frac{1}{4}\\)\u3068\u308f\u304b\u308a\u307e\u3057\u305f\u3002\u307e\u305f\u3001\u03b1\u3092\u5229\u7528\u3057\u3066\u03b2\u3092\u8a08\u7b97\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\((\u03b1,\u03b2)=(-1,-2)\\)<\/p>\n<p>\\((\u03b1,\u03b2)=\\left(\\displaystyle\\frac{1}{4},3\\right)\\)<\/p>\n<p>\u03b1\u3068\u03b2\u304c\u308f\u304b\u3063\u305f\u305f\u3081\u3001\\(a_{n+1}+\u03b1b_{n+1}=\u03b2(a_n+\u03b1b_n)\\)\u3078\u4ee3\u5165\u3059\u308b\u3068\u3001\u4ee5\u4e0b\u306e2\u3064\u306e\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_{n+1}-b_{n+1}=-2(a_n-b_n)\\)<\/li>\n<li>\\(a_{n+1}+\\displaystyle\\frac{1}{4}b_{n+1}=3\\left(a_n+\\displaystyle\\frac{1}{4}b_n\\right)\\)<\/li>\n<\/ul>\n<p>\\(a_{n+1}-b_{n+1}=-2(a_n-b_n)\\)\u306b\u3064\u3044\u3066\u3001\u6570\u5217\\(\\{a_n-b_n\\}\\)\u306f\u521d\u9805\\(a_1-b_1=-1\\)\u3001\u516c\u6bd4\\(-2\\)\u306e\u7b49\u6bd4\u6570\u5217\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_n-b_n=-(-2)^{n-1}\\) &#8211; \u2460<\/li>\n<\/ul>\n<p>\u307e\u305f\\(a_{n+1}+\\displaystyle\\frac{1}{4}b_{n+1}=3\\left(a_n+\\displaystyle\\frac{1}{4}b_n\\right)\\)\u306b\u3064\u3044\u3066\u3001\u6570\u5217\\(\\left\\{a_n+\\displaystyle\\frac{1}{4}b_n\\right\\}\\)\u306f\u521d\u9805\\(a_1+\\displaystyle\\frac{1}{4}b_1=\\displaystyle\\frac{3}{2}\\)\u3001\u516c\u6bd43\u306e\u7b49\u6bd4\u6570\u5217\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_n+\\displaystyle\\frac{1}{4}b_n=\\displaystyle\\frac{3}{2}\u00b73^{n-1}\\)\\(=\\displaystyle\\frac{3^n}{2}\\) &#8211; \u2461<\/li>\n<\/ul>\n<p>\u305d\u3053\u3067\u30012\u3064\u306e\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3067\\(a_n\\)\u3068\\(b_n\\)\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p><strong>\u30fb\\(\u2460+\u2461\u00d74\\)\u3088\u308a<\/strong><\/p>\n<p>\\(5a_n=-(-2)^{n-1}+2\u00b73^n\\)<\/p>\n<p>\\(a_n=\\displaystyle\\frac{-(-2)^{n-1}+2\u00b73^n}{5}\\)<\/p>\n<p><strong>\u30fb\\(\u2460-\u2461\\)\u3088\u308a<\/strong><\/p>\n<p>\\(-\\displaystyle\\frac{5}{4}b_n=-(-2)^{n-1}-\\displaystyle\\frac{3^n}{2}\\)<\/p>\n<p>\\(b_n=\\displaystyle\\frac{4\u00b7(-2)^{n-1}+2\u00b73^n}{5}\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u6570\u5217\\(\\{a_n\\}\\)\u3068\\(\\{b_n\\}\\)\u306e\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u03b1\u3068\u03b2\u306e\u4fc2\u6570\u3092\u5f97\u305f\u5f8c\u3001\u7b49\u6bd4\u6570\u5217\u578b\u306e\u6f38\u5316\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n<h3>\u96a3\u63a53\u9805\u9593\u306e\u6f38\u5316\u5f0f\u3078\u5909\u5f62\u3059\u308b<\/h3>\n<p>\u96a3\u63a53\u9805\u9593\u306e\u6f38\u5316\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u308c\u3070\u3001\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067\u3001<span style=\"color: #0000ff;\">\u9023\u7acb\u6f38\u5316\u5f0f\u306b\u5b58\u5728\u3059\u308b\\(a_n\\)\u307e\u305f\u306f\\(b_n\\)\u3092\u6d88\u53bb\u3059\u308b\u3053\u3068\u3067\u96a3\u63a53\u9805\u9593\u306e\u6f38\u5316\u5f0f\u3078\u5909\u5f62\u3057\u307e\u3057\u3087\u3046\u3002<\/span><\/p>\n<p>\u4f8b\u984c\u3068\u3057\u3066\u3001\u5148\u307b\u3069\u306e\u554f\u984c\u3092\u518d\u3073\u8a18\u3057\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_1=1\\)\u3001\\(b_1=2\\)\u3001\\(a_{n+1}=2a_n+b_n\\)\u3001\\(b_{n+1}=4a_n-b_n\\)\u306b\u3088\u3063\u3066\u5b9a\u3081\u3089\u308c\u308b\u6570\u5217\\(\\{a_n\\}\\)\u3068\\(\\{b_n\\}\\)\u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3057\u3087\u3046\u3002<\/li>\n<\/ul>\n<p>\u4ee5\u4e0b\u306e2\u3064\u306e\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u3067\\(a_n\\)\u307e\u305f\u306f\\(b_n\\)\u3092\u6d88\u53bb\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<ul>\n<li>\\(a_{n+1}=2a_n+b_n\\) &#8211; \u2460<\/li>\n<li>\\(b_{n+1}=4a_n-b_n\\) &#8211; \u2461<\/li>\n<\/ul>\n<p>\u307e\u305a\u3001\u2460\u3092\u5229\u7528\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3057\u307e\u3059\u3002<\/p>\n<p>\\(a_{n+1}=2a_n+b_n\\)<\/p>\n<p>\\(b_n=a_{n+1}-2a_n\\)<\/p>\n<p>\u307e\u305f\\(b_n=a_{n+1}-2a_n\\)\u306b\u3064\u3044\u3066\u3001\\(n\\)\u3092\\(n+1\\)\u3092\u5909\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<ul>\n<li>\\(b_{n+1}=a_{n+2}-2a_{n+1}\\)<\/li>\n<\/ul>\n<p>\u305d\u3053\u3067\u3001\\(b_n=a_{n+1}-2a_n\\)\u3068\\(b_{n+1}=a_{n+2}-2a_{n+1}\\)\u3092\u2461\u3078\u4ee3\u5165\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(b_{n+1}=4a_n-b_n\\)<\/p>\n<p>\\(a_{n+2}-2a_{n+1}=4a_n\\)\\(-(a_{n+1}-2a_n)\\)<\/p>\n<p>\\(a_{n+2}-a_{n+1}-6a_n=0\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u96a3\u63a53\u9805\u9593\u306e\u6f38\u5316\u5f0f\u3092\u4f5c\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u3042\u3068\u306f\u3001\u96a3\u63a53\u9805\u9593\u306e\u6f38\u5316\u5f0f\u306e\u89e3\u304d\u65b9\u3092\u77e5\u3063\u3066\u3044\u308c\u3070\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u7279\u6027\u65b9\u7a0b\u5f0f\\(x^2-x-6=0\\)\u3092\u89e3\u304f\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n<p>\\(x^2-x-6=0\\)<\/p>\n<p>\\((x+2)(x-3)=0\\)<\/p>\n<p>\\(x=-2,3\\)<\/p>\n<p>\\(x=-2,3\\)\u3067\u3042\u308b\u305f\u3081\u3001\u4ee5\u4e0b\u306e2\u3064\u306e\u5f0f\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_{n+2}-3a_{n+1}=-2(a_{n+1}-3a_n)\\)<\/li>\n<li>\\(a_{n+2}+2a_{n+1}=3(a_{n+1}+2a_n)\\)<\/li>\n<\/ul>\n<p>\\(a_{n+2}-3a_{n+1}=-2(a_{n+1}-3a_n)\\)\u306b\u7740\u76ee\u3059\u308b\u3068\u3001\u6570\u5217\\(\\{a_{n+1}-3a_n\\}\\)\u306f\u521d\u9805\\(a_2-3a_1=1\\)\u3001\u516c\u6bd4\\(-2\\)\u306e\u7b49\u6bd4\u6570\u5217\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_{n+1}-3a_n=(-2)^{n-1}\\) &#8211; \u2462<\/li>\n<\/ul>\n<p>\u307e\u305f\\(a_{n+2}+2a_{n+1}=3(a_{n+1}+2a_n)\\)\u306b\u7740\u76ee\u3059\u308b\u3068\u3001\u6570\u5217\\(\\{a_{n+1}+2a_n\\}\\)\u306f\u521d\u9805\\(a_2+2a_1=6\\)\u3001\u516c\u6bd43\u306e\u7b49\u6bd4\u6570\u5217\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<ul>\n<li>\\(a_{n+1}+2a_n=6\u00b73^{n-1}\\)\\(=2\u00b73^n\\) &#8211; \u2463<\/li>\n<\/ul>\n<p>\\(\u2463-\u2462\\)\u3088\u308a\u3001\u4ee5\u4e0b\u306e\u5f0f\u3092\u4f5c\u308c\u307e\u3059\u3002<\/p>\n<p>\\(5a_n=-(-2)^{n-1}+2\u00b73^n\\)<\/p>\n<p>\\(a_n=\\displaystyle\\frac{-(-2)^{n-1}+2\u00b73^n}{5}\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\u5148\u307b\u3069\u3068\u540c\u3058\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u307e\u305f\u5148\u307b\u3069\u8a08\u7b97\u3057\u305f\u901a\u308a\u3001\\(b_n=a_{n+1}-2a_n\\)\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u3053\u306e\u5f0f\u306b\\(a_n\\)\u3092\u4ee3\u5165\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\\(b_n=a_{n+1}-2a_n\\)<\/p>\n<p>\\(b_n=\\displaystyle\\frac{-(-2)^n+2\u00b73^{n+1}}{5}\\)\\(-2\u00b7\\displaystyle\\frac{-(-2)^{n-1}+2\u00b73^n}{5}\\)<\/p>\n<p>\\(b_n=\\displaystyle\\frac{2\u00b7(-2)^{n-1}+6\u00b73^n}{5}\\)\\(+\u00b7\\displaystyle\\frac{2(-2)^{n-1}-4\u00b73^n}{5}\\)<\/p>\n<p>\\(b_n=\\displaystyle\\frac{4\u00b7(-2)^{n-1}+2\u00b73^n}{5}\\)<\/p>\n<p>\u3053\u3046\u3057\u3066\u3001\\(b_n\\)\u306b\u3064\u3044\u3066\u3082\u5148\u307b\u3069\u3068\u540c\u3058\u7b54\u3048\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002\u3069\u3061\u3089\u306e\u65b9\u6cd5\u3092\u63a1\u7528\u3057\u3066\u3082\u554f\u984c\u306a\u304f\u3001\u89e3\u304d\u3084\u3059\u3044\u65b9\u6cd5\u3092\u5229\u7528\u3057\u3066\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n<h2>\u9023\u7acb\u6f38\u5316\u5f0f\u3092\u89e3\u304d\u3001\u4e00\u822c\u9805\u3092\u5f97\u308b<\/h2>\n<p>\u7279\u6b8a\u306a\u6f38\u5316\u5f0f\u306e\u4e00\u3064\u306b\u9023\u7acb\u6f38\u5316\u5f0f\u304c\u3042\u308a\u307e\u3059\u3002\u9023\u7acb\u6f38\u5316\u5f0f\u306e\u7b54\u3048\u3092\u5f97\u308b\u3068\u304d\u3001\u89e3\u304d\u65b9\u3092\u77e5\u3089\u306a\u3051\u308c\u3070\u3001\u4e00\u822c\u9805\u3092\u5f97\u308b\u306e\u306f\u96e3\u3057\u3044\u3067\u3059\u3002\u305d\u3053\u3067\u3001\u3069\u306e\u3088\u3046\u306b\u554f\u984c\u3092\u89e3\u3051\u3070\u3044\u3044\u306e\u304b\u5b66\u3073\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u30b7\u30f3\u30d7\u30eb\u306a\u9023\u7acb\u6f38\u5316\u5f0f\u3068\u3057\u3066\u306f\u3001\u5bfe\u79f0\u578b\u306e\u9023\u7acb\u6f38\u5316\u5f0f\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u3001\u8db3\u3057\u7b97\u3068\u5f15\u304d\u7b97\u306e\u4e21\u65b9\u3092\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u5f0f\u3092\u4f5c\u308a\u307e\u3057\u3087\u3046\u3002<\/p>\n<p>\u305f\u3060\u591a\u304f\u306e\u5834\u5408\u3001\u5bfe\u79f0\u578b\u306e\u9023\u7acb\u6f38\u5316\u5f0f\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u3053\u306e\u5834\u5408\u3001\u6f38\u5316\u5f0f\u3092\u7b49\u6bd4\u6570\u5217\u578b\u3078\u5909\u5f62\u3059\u308b\u3053\u3068\u30672\u3064\u306e\u5f0f\u3092\u4f5c\u308a\u307e\u3057\u3087\u3046\u3002\u307e\u305f\u306f\u3001\u9023\u7acb\u6f38\u5316\u5f0f\u3092\u5229\u7528\u3057\u3066\u96a3\u63a53\u9805\u9593\u306e\u6f38\u5316\u5f0f\u3078\u5909\u5f62\u3059\u308b\u3053\u3068\u3067\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002<\/p>\n<p>\u6f38\u5316\u5f0f\u306e\u554f\u984c\u3092\u89e3\u304f\u30b3\u30c4\u306f\u300c\u89e3\u304d\u65b9\u3092\u899a\u3048\u3066\u3044\u308b\u304b\u3069\u3046\u304b\u300d\u3060\u3051\u3067\u3059\u3002\u30d2\u30f3\u30c8\u306a\u3057\u306b\u7b54\u3048\u3092\u5f97\u308b\u306e\u306f\u96e3\u3057\u3044\u305f\u3081\u3001\u7b54\u3048\u3092\u5f97\u308b\u305f\u3081\u306e\u904e\u7a0b\u3092\u5b66\u3073\u307e\u3057\u3087\u3046\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u9023\u7acb\u6f38\u5316\u5f0f\u3067\u306f\u3001\u7570\u306a\u308b\u6f38\u5316\u5f0f\u3092\u5229\u7528\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u4e00\u822c\u9805\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u9023\u7acb\u6f38\u5316\u5f0f\u3067\u306f\u89e3\u304d\u65b9\u304c\u6c7a\u307e\u3063\u3066\u3044\u308b\u305f\u3081\u3001\u7b54\u3048\u3092\u5f97\u308b\u65b9\u6cd5\u3092\u899a\u3048\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002 \u306a\u304a\u3001\u9023\u7acb\u6f38\u5316\u5f0f\u306b\u306f3\u3064\u306e\u89e3\u304d\u65b9\u304c\u3042\u308a\u307e\u3059\u3002\u3069\u306e\u65b9\u6cd5\u3092\u5229\u7528\u3057 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":12130,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":{"0":"post-12115","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\/12115","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=12115"}],"version-history":[{"count":12,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/posts\/12115\/revisions"}],"predecessor-version":[{"id":12232,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/posts\/12115\/revisions\/12232"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/media\/12130"}],"wp:attachment":[{"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/media?parent=12115"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/categories?post=12115"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/hatsudy.com\/jp\/wp-json\/wp\/v2\/tags?post=12115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}