{"id":6221,"date":"2025-12-24T17:21:35","date_gmt":"2025-12-24T08:21:35","guid":{"rendered":"https:\/\/math-travel.com\/?p=6221"},"modified":"2026-02-11T17:30:18","modified_gmt":"2026-02-11T08:30:18","slug":"kaisasuuretu","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-b\/kaisasuuretu\/","title":{"rendered":"\u968e\u5dee\u6570\u5217\u306e\u4e00\u822c\u9805\u306e\u6c42\u3081\u65b9\uff1a\u516c\u5f0f\u306e\u4f7f\u3044\u65b9\u3068\u300cn\u22672\u300d\u306e\u6761\u4ef6\u304c\u5fc5\u8981\u306a\u7406\u7531"},"content":{"rendered":"\n

\u6570\u5b66B\u6570\u5217\u306b\u304a\u3044\u3066\u3001\u591a\u304f\u306e\u9ad8\u6821\u751f\u3092\u60a9\u307e\u305b\u308b\u306e\u304c\u300c\u968e\u5dee\u6570\u5217<\/span>\u300d\u3067\u3059\u3002<\/span><\/p>\n\n\n\n

\u4eca\u56de\u89e3\u6c7a\u3059\u308b\u60a9\u307f<\/span><\/div>
\n

\u300c\u968e\u5dee\u6570\u5217\u3063\u3066\u306a\u306b\uff1f\u300d <\/p>\n\n\n\n

\u300c\u968e\u5dee\u6570\u5217\u306e\u4e00\u822c\u9805\u306e\u6c42\u3081\u65b9\u306f\uff1f\u300d<\/p>\n\n\n\n

\u300c\u548c\u306e\u516c\u5f0f\u3092\u5fd8\u308c\u3066\u3057\u307e\u3063\u305f\u300d<\/p>\n<\/div><\/div>\n\n\n\n

\u4eca\u56de\u306f\u968e\u5dee\u6570\u5217\u306b\u95a2\u3059\u308b\u3053\u3093\u306a\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002<\/p>\n\n\n

\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u968e\u5dee\u6570\u5217\u306f\u5206\u304b\u308b\u3051\u3069\u3001\u4e00\u822c\u9805\u3068\u304b\u306b\u306a\u308b\u3068\u3088\u304f\u5206\u304b\u3089\u306a\u304f\u3066\u2026<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n

\n
\"\u968e\u5dee\u6570\u5217\"<\/figure>\n<\/div>\n\n\n

\u968e\u5dee\u6570\u5217\u3068\u306f\u300c\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u96a3\u63a5\u3059\u308b\u9805\u306e\u5dee\u3092\u9805\u3068\u3059\u308b\u6570\u5217<\/span>\u300d\u3092\u6307\u3057\u307e\u3059\u3002

\u968e\u5dee\u6570\u5217\u306f\u6570\u5217\u306e\u306a\u304b\u3067\u3082\u3001\u304b\u306a\u308a\u91cd\u8981\u306a\u6570\u5217\u306e1\u3064\u3067\u3059\u3002<\/span><\/p>\n\n\n\n

\u968e\u5dee\u6570\u5217\u3092\u3057\u3063\u304b\u308a\u7406\u89e3\u3057\u3066\u304a\u304f\u3068\u3001\u69d8\u3005\u306a\u6570\u5217\u306e\u554f\u984c\u306b\u5bfe\u5fdc\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u672c\u8a18\u4e8b\u3067\u306f\u968e\u5dee\u6570\u5217\u3092\u7528\u3044\u305f\u4e00\u822c\u9805\u3068\u548c\u306e\u6c42\u3081\u65b9\u306b\u3064\u3044\u3066\u89e3\u8aac<\/span>\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u6570\u5217\u304c\u82e6\u624b\u306a\u65b9\u3084\u3001\u3053\u308c\u304b\u3089\u6570\u5217\u3092\u5b66\u7fd2\u3059\u308b\u65b9\u306e\u53c2\u8003\u306b\u306a\u308b\u306e\u3067\u3001\u305c\u3072\u6700\u5f8c\u307e\u3067\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n

\"\"\u30b7\u30fc\u30bf<\/span><\/div>
\n

\u6c17\u306b\u306a\u308b\u898b\u51fa\u3057\u3092\u30af\u30ea\u30c3\u30af\u3057\u3066\u3001
\u305c\u3072\u6700\u5f8c\u307e\u3067\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u968e\u5dee\u6570\u5217\u3068\u306f\uff1f<\/h2>\n\n\n\n

\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u96a3\u308a\u5408\u30462\u3064\u306e\u9805\u306e\u5dee<\/p>\n\n\n\n

\\[b_{n}=a_{n+1}-a_{n}\u3000(n=1,2,3,\\cdots)\\]<\/p>\n\n\n\n

\u3092\u9805\u3068\u3059\u308b\u6570\u5217\\(\\{b_{n}\\}\\)\u3092\u3001\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u968e\u5dee\u6570\u5217\u3068\u3044\u3044\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u968e\u5dee\u6570\u5217\"<\/figure>\n<\/div>\n\n\n

\u4f8b\u3048\u3070\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u6570\u5217\u304c\u3042\u3063\u305f\u3068\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n

\n
\"\u968e\u5dee\u6570\u5217\u306e\u4f8b1\"<\/figure>\n<\/div>\n\n\n

\u3053\u306e\u3068\u304d\u306e\u968e\u5dee\u6570\u5217\u306f\u521d\u9805\u304c3\u3067\u30013\u305a\u3064\u5897\u52a0\u3057\u3066\u3044\u308b\u306e\u3067\u300c\u521d\u98053\u3001\u516c\u5dee3\u306e\u7b49\u5dee\u6570\u5217\u300d\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\u968e\u5dee\u6570\u5217\u306e\u4f8b\"<\/figure>\n<\/div>\n\n\n
\u968e\u5dee\u6570\u5217<\/span><\/div>
\n

\u968e\u5dee\u6570\u5217\u3000\u21d2\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u96a3\u63a5\u3059\u308b\u9805\u306e\u5dee\u3092\u9805\u3068\u3059\u308b\u6570\u5217<\/p>\n\n\n\n

\u521d\u9805\uff1a\u6700\u521d\u306e\u9805<\/p>\n\n\n\n

\u7b49\u5dee\u6570\u5217\uff1a\u4e00\u5b9a\u306e\u5dee\u3067\u5909\u5316\u3059\u308b\u6570\u5217<\/p>\n\n\n\n

\u7b49\u6bd4\u6570\u5217\uff1a\u4e00\u5b9a\u306e\u6bd4\u3067\u5909\u5316\u3059\u308b\u6570\u5217<\/p>\n<\/div><\/div>\n\n\n\n

<\/span><\/p>\n\n\n\n

\u95a2\u9023\u8a18\u4e8b<\/span><\/div>
\n

\u7b49\u5dee\u6570\u5217\u306e\u516c\u5f0f\u307e\u3068\u3081\uff01\u4e00\u822c\u9805\u3068\u548c\u306e\u516c\u5f0f\u3092\u5206\u304b\u308a\u3084\u3059\u304f\u89e3\u8aac\uff01<\/p>\n\n\n\n

\u7b49\u6bd4\u6570\u5217\u306e\u516c\u5f0f\u307e\u3068\u3081\uff01\u4e00\u822c\u9805\u3068\u548c\u306e\u516c\u5f0f\u3092\u5206\u304b\u308a\u3084\u3059\u304f\u89e3\u8aac\uff01<\/p>\n<\/div><\/div>\n\n\n\n

\u968e\u5dee\u6570\u5217\u3068\u4e00\u822c\u9805<\/h2>\n\n\n\n

\u968e\u5dee\u6570\u5217\u3092\u7528\u3044\u305f\u4e00\u822c\u9805\u306e\u516c\u5f0f<\/h3>\n\n\n\n

\u4e00\u822c\u9805\u3068\u306f\u3001\u6570\u5217\u306e\u7b2c\\(n\\)\u9805\\(a_{n}\\)\u3092\\(n\\)\u3084\u6570\u5b57\u3092\u7528\u3044\u3066\u8868\u3057\u305f\u3082\u306e\u3067\u3059\u3002<\/p>\n\n\n\n

\u968e\u5dee\u6570\u5217\u3092\u7528\u3044\u3066\u3001\u3082\u3068\u306e\u6570\u5217\u306e\u4e00\u822c\u9805\u3092\u8868\u3059\u3068\u4ee5\u4e0b\u306e\u516c\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u968e\u5dee\u6570\u5217\u3068\u4e00\u822c\u9805<\/span><\/div>
\n

\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u968e\u5dee\u6570\u5217\u3092\\(\\{b_{n}\\}\\)\u3068\u3059\u308b\u3068\u3001n\u22672\u306e\u3068\u304d<\/p>\n\n\n\n

\\[a_{n}=a_{1}+\\sum_{k=1}^{n-1} b_{k}\\]<\/p>\n<\/div><\/div>\n\n\n

\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u516c\u5f0f\u306b\u5909\u306a\u8a18\u53f7\u304c\u5165\u3063\u3066\u3044\u308b\u3088<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n

\"\"\u30b7\u30fc\u30bf<\/span><\/div>
\n

\u03a3\uff08\u30b7\u30b0\u30de\uff09\u3092\u77e5\u3089\u306a\u3044\u4eba\u306f\u5148\u306b\u30b7\u30b0\u30de\u306b\u3064\u3044\u3066\u306e\u8a18\u4e8b\u3092\u8aad\u3093\u3067\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u968e\u5dee\u6570\u5217\u3092\u7528\u3044\u305f\u4e00\u822c\u9805\u306e\u8a3c\u660e<\/h3>\n\n\n\n

\u306a\u305c\u03a3\uff08\u30b7\u30b0\u30de\uff09\u3092\u4f7f\u3046\u306e\u3067\u3057\u3087\u3046\u304b\uff1f<\/p>\n\n\n\n

\u6570\u5217\u306e\u521d\u9805\\(a_{1}\\)\u306b\u5bfe\u3057\u3066\u3001\u968e\u5dee\u6570\u5217\\(\\{b_{n}\\}\\)\u3092\u52a0\u3048\u3066\u3044\u304f\u3068\u8003\u3048\u307e\u3059\u3002<\/p>\n\n\n\n

\u6570\u5217\\(a_{1}\u3000a_{2}\u3000a_{3}\u3000a_{4} …\u3000a_{n}\\)\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u306e\u6570\u5217\u306e\u968e\u5dee\u6570\u5217\u3092\\(\\{b_{n}\\}\\)\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n

\n

\\(a_{1}=a_{1}\\)\u2190\u521d\u9805<\/p>\n\n\n\n

\\(a_{2}=a_{1} + b_{1}\\)<\/p>\n\n\n\n

\\(a_{3}=a_{1} + b_{1}+b_{2}\\)<\/p>\n\n\n\n

\\(a_{4}=a_{1} + b_{1}+b_{2}+b_{3}\\)<\/p>\n<\/div><\/div>\n\n\n\n

\u3053\u306e\u3088\u3046\u306b\u521d\u9805\\(a_{1}\\)\u306b\u968e\u5dee\u6570\u5217\u306e\u9805\u3092\u52a0\u3048\u3066\u3044\u304f\u306e\u3067\u3001<\/p>\n\n\n\n

\n

\\(a_{1}=a_{1}\\)<\/p>\n\n\n\n

\\(a_{2}=a_{1} + b_{1}\\)<\/p>\n\n\n\n

\\(a_{3}=a_{1} + b_{1}+b_{2}\\)<\/p>\n\n\n\n

\uff1a<\/p>\n\n\n\n

\\(a_{n}=a_{1} + b_{1}+b_{2}+ \\cdots +b_{n-1} \\)<\/p>\n<\/div><\/div>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u3001\u968e\u5dee\u6570\u5217\u3092\u7528\u3044\u305f\u4e00\u822c\u9805\u306f<\/p>\n\n\n\n

\\[a_{n}=a_{1}+\\sum_{k=1}^{n-1} b_{k}\\]<\/p>\n\n\n\n

\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n

\"\"\u30b7\u30fc\u30bf<\/span><\/div>
\n

\u5177\u4f53\u7684\u306a\u6570\u5b57\u3092\u4f7f\u3063\u3066\u4e00\u822c\u9805\u3092\u8003\u3048\u3066\u307f\u3088\u3046\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u968e\u5dee\u6570\u5217\u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u308b<\/h3>\n\n\n\n

\u3053\u3053\u306b\u6570\u5217\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

3 , 5 , 8 , 12 , 17 , 23 \u2026<\/p>\n\n\n\n

\u3053\u306e\u6570\u5217\u3060\u3051\u3067\u306f\u6cd5\u5247\u304c\u898b\u3048\u307e\u305b\u3093\u304c\u3001\u5404\u9805\u306e\u5dee\u3092\u6c42\u3081\u308b\u3068<\/p>\n\n\n\n

2 , 3 , 4 , 5 , 6 \u2026<\/p>\n\n\n\n

\u968e\u5dee\u6570\u5217\u304c\u300c\u521d\u98052\u3001\u516c\u5dee1\u306e\u7b49\u5dee\u6570\u5217\u300d\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u3053\u3067\u968e\u5dee\u6570\u5217\u3092\u7528\u3044\u305f\u6570\u5217\u306e\u4e00\u822c\u9805\u306e\u516c\u5f0f\u3092\u601d\u3044\u51fa\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u968e\u5dee\u6570\u5217\u3068\u4e00\u822c\u9805<\/span><\/div>
\n

\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u968e\u5dee\u6570\u5217\u3092\\(\\{b_{n}\\}\\)\u3068\u3059\u308b\u3068\u3001n\u22672\u306e\u3068\u304d<\/p>\n\n\n\n

\\[a_{n}=a_{1}+\\sum_{k=1}^{n-1} b_{k}\\]<\/p>\n<\/div><\/div>\n\n\n\n

\u3082\u3068\u306e\u6570\u5217\\(\\{a_{n}\\}\\)\u306f\u521d\u98053\u3067\u3001\u968e\u5dee\u6570\u5217\\(\\{b_{n}\\}\\)\u306f\u300c\u521d\u98052\u3001\u516c\u5dee1\u306e\u7b49\u5dee\u6570\u5217\u300d\u306a\u306e\u3067<\/p>\n\n\n\n

\\[b_{n}=2+(n-1)\\]<\/p>\n\n\n\n

\\(n\u22672\\)\u306e\u3068\u304d\u3001<\/p>\n\n\n\n

\\begin{eqnarray}
a_{n}&=&a_{1}+\\sum_{k=1}^{n-1} b_{k}\\\\
&=&3+\\sum_{k=1}^{n-1} \\{2+(k-1)\\}\\\\
&=&3+\\sum_{k=1}^{n-1} (1+k)\\\\
&=&3+\\sum_{k=1}^{n-1} 1+\\sum_{k=1}^{n-1} k
\\end{eqnarray}<\/p>\n\n\n\n

\\begin{eqnarray}
a_{n}&=&3+\\sum_{k=1}^{n-1} 1+\\sum_{k=1}^{n-1} k\\\\
\\displaystyle &=&3+(n-1)+\\frac{1}{2}(n-1)n\\\\
\\displaystyle &=&\\frac{1}{2}n^{2}+\\frac{1}{2}n+2\\\\
\\end{eqnarray}<\/p>\n\n\n\n

\u3088\u3063\u3066\u3001\\[\\displaystyle a_{n}=\\frac{1}{2}n^{2}+\\frac{1}{2}n+2\\]<\/p>\n\n\n\n

\\(n=1\\)\u306e\u3068\u304d\\(a_{1}=3\\)\u3068\u306a\u308a\u3001\u521d\u9805\u3067\u3082\u6210\u308a\u7acb\u3064\u3002<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u3001\u4e0e\u3048\u3089\u308c\u305f\u6570\u5217\u306e\u4e00\u822c\u9805\u306f<\/p>\n\n\n\n

\\[\\displaystyle a_{n}=\\frac{1}{2}n^{2}+\\frac{1}{2}n+2\\]<\/p>\n\n\n

\"\"\u30b7\u30fc\u30bf<\/span><\/div>
\n

\u5fc3\u914d\u306a\u3068\u304d\u306f\u4ee3\u5165\u3057\u3066\u78ba\u304b\u3081\u3066\u307f\u3088\u3046\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u4e00\u822c\u9805\u304c\u4e0d\u5b89\u306a\u3068\u304d\u306f\u6570\u5b57\u3092\u4ee3\u5165\u3057\u3066\u78ba\u304b\u3081\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\u4e00\u822c\u9805\u306b\\(n=2\\)\u3092\u4ee3\u5165\u3057\u3066<\/p>\n\n\n\n

\\begin{eqnarray}
\\displaystyle a_{2}&=&\\frac{1}{2}\\cdot 2^{2}+\\frac{1}{2} \\cdot 2+2\\\\
&=&2+1+2\\\\
&=&5
\\end{eqnarray}<\/p>\n\n\n\n

\u3061\u3083\u3093\u3068\u540c\u3058\u5024\u306b\u306a\u3063\u305f\u306e\u3067\u9593\u9055\u3044\u306a\u3055\u305d\u3046\u3067\u3059\u3002<\/p>\n\n\n\n

\u968e\u5dee\u6570\u5217\u306e\u548c<\/h2>\n\n\n\n

\u968e\u5dee\u6570\u5217\u306e\u548c<\/h3>\n\n\n\n

\u6570\u5217\u306e\u9805\u3092\u8db3\u3059\u3053\u3068\u3092\u6570\u5217\u306e\u548c\u3068\u3044\u3044\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\u6570\u5217{3 , 5 , 8 , 11}\u306e\u521d\u9805\u304b\u3089\u7b2c4\u9805\u307e\u3067\u306e\u548c\u306f27\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

3+5+8+11=27<\/p>\n\n\n\n

\u3053\u306e\u3088\u3046\u306b\u6570\u5217\u306e\u548c\u3092\u6c42\u3081\u308b\u554f\u984c\u306f\u3088\u304f\u51fa\u984c\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

\u3082\u3068\u306e\u6570\u5217\u304c\u7b49\u5dee\u6570\u5217\u304b\u7b49\u6bd4\u6570\u5217\u306e\u3068\u304d\u306f\u548c\u306e\u516c\u5f0f\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f<\/span><\/div>
\n

\u521d\u9805\\(a\\)\u3001\u516c\u5dee\\(d\\)\u3001\u672b\u9805\\(l\\)\u3001\u9805\u6570\\(n\\)\u306e\u7b49\u5dee\u6570\u5217\u306e\u548c\u3092\\(S_{n}\\)\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n

\\begin{eqnarray}
\\displaystyle S_{n}&=&\\frac{n}{2}(a+l)\\\\
\\displaystyle &=&\\frac{n}{2}\\{2a+(n-1)d\\}
\\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

\u4eca\u56de\u306f\u968e\u5dee\u6570\u5217\u3067\u3059\u304c\u3001\u4e0a\u8a18\u306e\u516c\u5f0f\u3082\u78ba\u5b9f\u306b\u899a\u3048\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/span><\/p>\n\n\n\n

\u968e\u5dee\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f<\/span><\/div>
\n

\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u521d\u9805\u304b\u3089\u7b2c\\(n\\)\u9805\u307e\u3067\u306e\u548c\u3092\\(S_{n}\\)\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n

\\[\\displaystyle S_{n}=\\sum_{k=1}^{n} a_{k}\\]<\/p>\n<\/div><\/div>\n\n\n\n

\u5b9f\u969b\u306b\u6570\u5b57\u3092\u4f7f\u3063\u3066\u548c\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u968e\u5dee\u6570\u5217\u306e\u548c\u3092\u6c42\u3081\u308b<\/h3>\n\n\n\n

\u3053\u306e\u3088\u3046\u306a\u6570\u5217\u304c\u3042\u3063\u305f\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

1 , 3 , 7 , 13 , 21 , 31 , 43 \u2026<\/p>\n\n\n\n

\u3053\u306e\u6570\u5217\u306e\u968e\u5dee\u6570\u5217\u306f\u300c\u521d\u98052\u3001\u516c\u5dee2\u306e\u7b49\u5dee\u6570\u5217\u300d\u3067\u3059\u3002<\/p>\n\n\n

\n
\"\u968e\u5dee\u6570\u5217\"<\/figure>\n<\/div>\n\n\n

\u6570\u5217\u306e\u548c\u3092\u6c42\u3081\u308b\u306b\u306f\u3001\u307e\u305a\u4e00\u822c\u9805\\(\\{a_{n}\\}\\)\u3092\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u3053\u3067\u4e00\u822c\u9805\u3092\u6c42\u3081\u308b\u516c\u5f0f\u3092\u601d\u3044\u51fa\u3057\u3066\u3001<\/p>\n\n\n\n

\\begin{eqnarray}
a_{n}&=&1+\\sum_{k=1}^{n-1} \\{2+2(k-1)\\}\\\\
&=&1+\\sum_{k=1}^{n-1} 2k\\\\
\\displaystyle &=&1+2 \\cdot \\frac{1}{2}(n-1)n\\\\
&=&n^{2}-n+1
\\end{eqnarray}<\/p>\n\n\n

\"\"\u30b7\u30fc\u30bf<\/span><\/div>
\n

\u3053\u308c\u3067({a_{n}})\u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u306d\u3002<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u4e00\u822c\u9805\u304c\u5206\u304b\u308c\u3070\u3001\u6570\u5217\u306e\u548c\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u521d\u9805\\(\\{a_{n}\\}\\)\u304b\u3089\u7b2c\\(n\\)\u9805\\(\\{a_{n}\\}\\)\u307e\u3067\u306e\u548c\u306f\u3001<\/p>\n\n\n\n

\\begin{eqnarray}
S_{n}&=&\\sum_{k=1}^{n} a_{k}\\\\
&=&\\sum_{k=1}^{n} (n^{2}-n+1)\\\\
&=&\\sum_{k=1}^{n} n^{2}-\\sum_{k=1}^{n} n+\\sum_{k=1}^{n} 1\\\\
\\displaystyle &=&\\frac{1}{6}n(n+1)(2n+1)-\\frac{1}{2}n(n+1)+n\\\\
\\displaystyle &=&\\frac{1}{3}n(n^{2}+2)
\\end{eqnarray}<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u3001\u4e0e\u3048\u3089\u308c\u305f\u6570\u5217\u306e\u521d\u9805\u304b\u3089\u7b2c\\(n\\)\u9805\u307e\u3067\u306e\u548c\\(S_{n}\\)\u306f<\/p>\n\n\n\n

\\[\\displaystyle S_{n}=\\frac{1}{3}n(n^{2}+2)\\]<\/p>\n\n\n

\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u7b49\u5dee\u6570\u5217\u3084\u7b49\u6bd4\u6570\u5217\u306e\u3088\u3046\u306a\u548c\u306e\u516c\u5f0f\u306f\u306a\u3044\u3093\u3060\u306d<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n

\"\"\u30b7\u30fc\u30bf<\/span><\/div>
\n

\u305d\u3046\u306a\u3093\u3060\u3088\u3001\u8a08\u7b97\u304c\u5927\u5909\u3060\u3051\u3069\u03a3\u306e\u8a08\u7b97\u3092\u3059\u308b\u3057\u304b\u306a\u3044\u306d<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u968e\u5dee\u6570\u5217\u306e\u6f38\u5316\u5f0f<\/h2>\n\n\n\n

\u6f38\u5316\u5f0f\u304c\\(a_{n+1}=a_{n}+(n\u306e\u5f0f)\\)\u306e\u5f62\u306e\u3068\u304d\u3001\u968e\u5dee\u6570\u5217\u3092\u5229\u7528\u3057\u3066\u4e00\u822c\u9805\u3092\u6c42\u3081\u307e\u3059\u3002<\/p>\n\n\n\n

\u968e\u5dee\u6570\u5217\u3092\u5229\u7528\u3059\u308b\u6f38\u5316\u5f0f<\/span><\/div>
\n

\u6570\u5217\u306b\u304a\u3051\u308b\u7b2c\\(n\\)\u9805\u3092\\(a_{n}\\)\u3068\u3059\u308b\u3068\u304d<\/p>\n\n\n\n

\\[a_{n+1}=a_{n}+f(n)\\]<\/p>\n\n\n\n

\u5f0f\u5909\u5f62\u3092\u3059\u308b\u3068<\/p>\n\n\n\n

\\[a_{n+1}-a_{n}=f(n)\\]<\/p>\n\n\n\n

\u3053\u308c\u306f\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u968e\u5dee\u6570\u5217\u306e\u7b2c\\(n\\)\u9805\u304c\\(f(x)\\)\u3067\u3042\u308b\u3053\u3068\u3092\u8868\u3059\u3002<\/p>\n<\/div><\/div>\n\n\n\n

\u5b9f\u969b\u306b\u6f38\u5316\u5f0f\u304b\u3089\u4e00\u822c\u9805\u3092\u6c42\u3081\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u4f8b\u984c1<\/span><\/div>
\n

\u6b21\u306e\u6761\u4ef6\u306b\u3088\u3063\u3066\u5b9a\u3081\u3089\u308c\u308b\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u3088\u3046\u3002<\/p>\n\n\n\n

\\[a_{1}=3\u3000a_{n+1}=a_{n}+2^{n}\\]<\/p>\n<\/div><\/div>\n\n\n\n

\u6761\u4ef6\u3088\u308a\\[a_{n+1}-a_{n}=2^{n}\\]<\/p>\n\n\n\n

\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u968e\u5dee\u6570\u5217\u306e\u7b2c\\(n\\)\u9805\u304c\\(2^{n}\\)\u306a\u306e\u3067\u3001<\/p>\n\n\n\n

\\(n\u22672\\)\u306e\u3068\u304d<\/p>\n\n\n\n

\\begin{eqnarray}
a_{n}&=&a_{1}+\\sum_{k=1}^{n-1} 2^{k}\\\\
\\displaystyle &=&3 + \\frac{2(2^{n-1}-1)}{2-1}
\\end{eqnarray}<\/p>\n\n\n\n

\u3088\u3063\u3066\u3001<\/p>\n\n\n\n

\\[a_{n}=2^{n}+1\\]<\/p>\n\n\n\n

\\(n=1\\)\u306e\u3068\u304d\u3001\\(a_{1}=3\\)\u3068\u306a\u308a\u521d\u9805\u3067\u3082\u6210\u308a\u7acb\u3064\u3002<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u3001\u4e00\u822c\u9805\u306f\\(a_{n}=2^{n}+1\\)<\/p>\n\n\n\n

\u305d\u306e\u4ed6\u306e\u6570\u5217<\/h2>\n\n\n\n

\u4eca\u56de\u306f\u968e\u5dee\u6570\u5217\u3092\u30e1\u30a4\u30f3\u306b\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u6570\u5217\u306b\u306f\u968e\u5dee\u6570\u5217\u306e\u4ed6\u306b\u3082\u91cd\u8981\u306a\u6570\u5217\u304c\u3042\u308a\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\n
    \n
  • \u7b49\u5dee\u6570\u5217<\/li>\n\n\n\n
  • \u7b49\u6bd4\u6570\u5217<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n

    \u30fb\u7b49\u5dee\u6570\u5217<\/span><\/p>\n\n\n\n

    \u7b49\u5dee\u6570\u5217\u3068\u306f\u3001\u300c\u4e00\u5b9a\u306e\u5dee\u3067\u5909\u5316\u3059\u308b\u6570\u5217\u300d<\/span>\u3092\u6307\u3057\u307e\u3059\u3002<\/p>\n\n\n

    \n
    \"\u7b49\u5dee\u6570\u5217\"<\/figure>\n<\/div>\n\n\n

    \u7b49\u5dee\u6570\u5217\u306b\u3064\u3044\u3066\u306f\u3001\u300c\u7b49\u5dee\u6570\u5217\u306e\u516c\u5f0f\u307e\u3068\u3081\uff01\u4e00\u822c\u9805\u3068\u548c\u306e\u516c\u5f0f\u3092\u5206\u304b\u308a\u3084\u3059\u304f\u89e3\u8aac\uff01\u300d\u3067\u8a73\u3057\u304f\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

    \u30fb\u7b49\u6bd4\u6570\u5217<\/span><\/p>\n\n\n\n

    \u7b49\u6bd4\u6570\u5217\u3068\u306f\u3001\u300c\u521d\u3081\u306e\u9805\u306b\u4e00\u5b9a\u306e\u6570\u3092\u304b\u3051\u7d9a\u3051\u3066\u3044\u304f\u6570\u5217\u300d<\/span>\u3092\u6307\u3057\u307e\u3059\u3002<\/p>\n\n\n

    \n
    \"\u7b49\u6bd4\u6570\u5217\"<\/figure>\n<\/div>\n\n\n

    \u7b49\u6bd4\u6570\u5217\u306b\u3064\u3044\u3066\u306f\u3001\u300c\u7b49\u6bd4\u6570\u5217\u306e\u516c\u5f0f\u307e\u3068\u3081\uff01\u4e00\u822c\u9805\u3068\u548c\u306e\u516c\u5f0f\u3092\u5206\u304b\u308a\u3084\u3059\u304f\u89e3\u8aac\uff01\u300d\u3067\u8a73\u3057\u304f\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

    \u968e\u5dee\u6570\u5217\u300a\u7df4\u7fd2\u554f\u984c\u300b<\/h2>\n\n\n\n

    \u3053\u3053\u307e\u3067\u6570\u5217\u306e\u4e00\u822c\u9805\u306e\u6c42\u3081\u65b9\u3084\u548c\u306e\u6c42\u3081\u65b9\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u3066\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

    \u5b9f\u969b\u306b\u554f\u984c\u3092\u89e3\u3044\u3066\u307f\u3066\u3001\u8a08\u7b97\u306b\u6163\u308c\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

    \u7df4\u7fd2\u554f\u984c1<\/span><\/div>
    \n

    \u6b21\u306e\u6570\u5217\u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u3088\u3046\u3002<\/p>\n\n\n\n

    1 , 4 , 9 , 16 , 25 , 36 \u2026<\/p>\n<\/div><\/div>\n\n\n\n

    \n
    \u89e3\u7b54<\/span><\/i><\/i><\/span><\/summary>
    \n

    \u4e0e\u3048\u3089\u308c\u305f\u6570\u5217\u306e\u968e\u5dee\u6570\u5217\u306f\u300c\u521d\u98053\u3001\u516c\u5dee2\u306e\u7b49\u5dee\u6570\u5217\u300d\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

    \u3088\u3063\u3066\u3001<\/p>\n\n\n\n

    \\[b_{n}=3+2(n-1)=2n+1\\]<\/p>\n\n\n\n

    \u521d\u9805\\(a_{1}\\)\u3068\u968e\u5dee\u6570\u5217\\(\\{b_{n}\\}\\)\u304c\u5206\u304b\u3063\u305f\u306e\u3067\u4e00\u822c\u9805\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

    \\(n\u22672\\)\u306e\u3068\u304d<\/p>\n\n\n\n

    \\begin{eqnarray}
    a_{n}&=&1+\\sum_{k=1}^{n-1} (2k+1)\\\\
    \\displaystyle &=&1+2\\sum_{k=1}^{n-1} k +\\sum_{k=1}^{n-1} 1\\\\
    \\displaystyle &=&1+2 \\cdot \\frac{1}{2}(n-1)n +(n-1)\\\\
    &=&n^{2}
    \\end{eqnarray}<\/p>\n\n\n\n

    \u3059\u306a\u308f\u3061\\[a_{n}=n^{2}\\]<\/p>\n\n\n\n

    \\(n=1\\)\u306e\u3068\u304d\\(a_{1}=1\\)\u3068\u306a\u308a\u3001\u521d\u9805\u3067\u3082\u6210\u308a\u7acb\u3064\u3002<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001\u4e00\u822c\u9805\u306f\\[a_{n}=n^{2}\\]<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n

    \u7df4\u7fd2\u554f\u984c2<\/span><\/div>
    \n

    \u6b21\u306e\u6570\u5217\u306b\u304a\u3044\u3066\u3001\u521d\u9805\u304b\u3089\u7b2c\\(n\\)\u9805\u307e\u3067\u306e\u548c\u3092\u6c42\u3081\u3088\u3046\u3002
    1\u30004\u300013\u300040\u3000121\u3000\u2026<\/p>\n<\/div><\/div>\n\n\n\n

    \n
    \u89e3\u7b54<\/span><\/i><\/i><\/span><\/summary>
    \n

    \u307e\u305a\u306f\u4e00\u822c\u9805\\(\\{a_{n}\\}\\)\u3092\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002<\/p>\n\n\n\n

    \u4e0e\u3048\u3089\u308c\u305f\u6570\u5217\u306e\u968e\u5dee\u6570\u5217\u306f\u300c\u521d\u98053\u3001\u516c\u6bd43\u306e\u7b49\u6bd4\u6570\u5217\u300d\u306a\u306e\u3067\u3001<\/p>\n\n\n\n

    \\[b_{n}=3 \\cdot 3^{n-1}=3^{n}\\]<\/p>\n\n\n\n

    \\(n\u22672\\)\u306e\u3068\u304d<\/p>\n\n\n\n

    \\begin{eqnarray}
    a_{n}&=&1+\\sum_{k=1}^{n-1} 3^{k}\\\\
    \\displaystyle &=&1+\\frac{3(3^{n-1}-1)}{3-1}\\\\
    \\displaystyle &=&1+\\frac{1}{2}(3^{n}-3)\\\\
    \\displaystyle &=&\\frac{1}{2}(3^{n}-1)
    \\end{eqnarray}<\/p>\n\n\n\n

    \\(n=1\\)\u306e\u3068\u304d\\(a_{1}=1\\)\u3068\u306a\u308a\u3001\u521d\u9805\u3067\u3082\u6210\u308a\u7acb\u3064\u3002<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001\u4e00\u822c\u9805\u306f\\[\\displaystyle a_{n}=\\frac{1}{2}(3^{n}-1)\\]<\/p>\n\n\n\n

    \u3053\u308c\u3067\u6570\u5217\u306e\u548c\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

    \u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u521d\u9805\u304b\u3089\u7b2c\\(n\\)\u9805\u307e\u3067\u306e\u548c\\(S_{n}\\)\u306f<\/p>\n\n\n\n

    \\begin{eqnarray}
    S_{n}&=&\\sum_{k=1}^{n} a_{k}\\\\
    \\displaystyle &=&\\frac{1}{2} \\sum_{k=1}^{n} 3^{k}-\\frac{1}{2} \\sum_{k=1}^{n} 1\\\\
    \\displaystyle &=&\\frac{1}{2} \\cdot \\frac{3(3^{n}-1)}{3-1}-\\frac{1}{2} \\cdot n\\\\
    \\displaystyle &=&\\frac{3}{4}(3^{n}-1)-\\frac{1}{2}n\\\\
    \\end{eqnarray}<\/p>\n\n\n\n

    \u3057\u305f\u304c\u3063\u3066\u3001\u6c42\u3081\u308b\u6570\u5217\u306e\u548c\\(S_{n}\\)\u306f<\/p>\n\n\n\n

    \\[\\displaystyle S_{n}=\\frac{3}{4}(3^{n}-1)-\\frac{1}{2}n\\]<\/p>\n<\/div><\/details>\n<\/div>\n\n\n\n

    \u968e\u5dee\u6570\u5217\u307e\u3068\u3081<\/h2>\n\n\n\n

    \u4eca\u56de\u306f\u968e\u5dee\u6570\u5217\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002<\/span><\/p>\n\n\n\n

    \u968e\u5dee\u6570\u5217\u3068\u306f
    <\/span>\u21d2\u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u96a3\u63a5\u3059\u308b\u9805\u306e\u5dee\u3092\u9805\u3068\u3059\u308b\u6570\u5217<\/p>\n\n\n\n

    \u968e\u5dee\u6570\u5217\u3068\u4e00\u822c\u9805<\/span><\/div>
    \n

    \u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u968e\u5dee\u6570\u5217\u3092\\(\\{b_{n}\\}\\)\u3068\u3059\u308b\u3068\u3001n\u22672\u306e\u3068\u304d<\/p>\n\n\n\n

    \\[a_{n}=a_{1}+\\sum_{k=1}^{n-1} b_{k}\\]<\/p>\n<\/div><\/div>\n\n\n\n

    \u968e\u5dee\u6570\u5217\u306e\u548c\u306e\u516c\u5f0f<\/span><\/div>
    \n

    \u6570\u5217\\(\\{a_{n}\\}\\)\u306e\u521d\u9805\u304b\u3089\u7b2c\\(n\\)\u9805\u307e\u3067\u306e\u548c\u3092\\(S_{n}\\)\u3068\u3059\u308b\u3068\u3001<\/p>\n\n\n\n

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