\u4ee5\u4e0b\u306e\u30eb\u30fc\u30eb\u306b\u5f93\u3063\u3066, $5$\u4eba\u3067$1$\u56de\u3058\u3083\u3093\u3051\u3093\u3092\u884c\u3046\u3002\u3000\u2460$1$\u7a2e\u985e\u306e\u624b\u304c\u51fa\u305f\u3068\u304d, \u3042\u3044\u3053\u3068\u3059\u308b\u3002
\n\u2461$2$\u7a2e\u985e\u306e\u624b\u304c\u51fa\u305f\u3068\u304d,\u901a\u5e38\u306e\u3058\u3083\u3093\u3051\u3093\u306e\u30eb\u30fc\u30eb\u306b\u5f93\u3046\u3002
\n\u2462$3$\u7a2e\u985e\u306e\u624b\u304c\u51fa\u3066\u3044\u308b\u3068\u304d, \u540c\u3058\u624b\u3092\u51fa\u3057\u3066\u3044\u308b\u4eba\u304c\u5c11\u306a\u3044\u4eba\u3092\u52dd\u8005\u3068\u3059\u308b\u3002
\n\u4f8b\u3048\u3070, $2$\u4eba\u304c\u30b0\u30fc, $2$\u4eba\u304c\u30d1\u30fc, $1$\u4eba\u304c\u30c1\u30e7\u30ad\u3060\u3063\u305f\u5834\u5408, \u30c1\u30e7\u30ad\u3092\u51fa\u3057\u3066\u3044\u308b\u4eba\u304c\u52dd\u8005\u3068\u306a\u308a, $3$\u4eba\u304c\u30b0\u30fc, $1$\u4eba\u304c\u30d1\u30fc, $1$\u4eba\u304c\u30c1\u30e7\u30ad\u3060\u3063\u305f\u5834\u5408, \u30d1\u30fc\u3092\u51fa\u3057\u3066\u3044\u308b\u4eba\u3068\u3068\u30c1\u30e7\u30ad\u3092\u51fa\u3057\u3066\u3044\u308b\u4eba\u304c\u52dd\u8005\u3068\u306a\u308b\u3002
\n(1)\u3042\u3044\u3053\u3068\u306a\u308b\u78ba\u7387\u3092\u6c42\u3081\u3088\u3002
\n(2)$1$\u4eba\u306e\u52dd\u8005\u304c\u6c7a\u307e\u308b\u78ba\u7387\u3092\u6c42\u3081\u3088\u3002
\n(3)\u52dd\u8005\u306e\u4eba\u6570\u306e\u671f\u5f85\u5024\u3092\u6c42\u3081\u3088\u3002<\/div>\n<\/div>\n
(1),(2)\u3067\u5177\u4f53\u7684\u306a\u8a08\u7b97\u3092\u3057,\u4eca\u56de\u306e\u30eb\u30fc\u30eb\u306e\u78ba\u8a8d\u3057\u3066\u6700\u7d42\u76ee\u6a19\u306f(3)\u306e\u671f\u5f85\u5024\u306e\u8a08\u7b97\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\u89e3\u7b54\u306f\u3053\u3061\u3089<\/summary>\n(1)\u3092\u89e3\u304f\u3068\u304d,\u3042\u3044\u3053\u306b\u306a\u308b\u6761\u4ef6\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002\u4eca\u56de\u306e\u8a2d\u5b9a\u3067\u306f,\u3042\u3044\u3053\u306b\u306a\u308b\u306e\u306f\u30eb\u30fc\u30eb\u2460\u306e\u3068\u304d\u306b\u9650\u308a\u307e\u3059\u3002
\n$5$\u4eba\u306e\u624b\u306e\u51fa\u3057\u65b9\u306f$3^5$\u901a\u308a\u3067, \u3053\u306e\u3046\u3061\u3042\u3044\u3053\u3068\u306a\u308b\u306e\u306f,\u5168\u54e1\u304c\u30b0\u30fc\u307e\u305f\u306f\u30c1\u30e7\u30ad\u307e\u305f\u306f\u30d1\u30fc\u306e$3$\u901a\u308a\u306a\u306e\u3067\u78ba\u7387\u306f$$\\cfrac{3}{3^5}=\\cfrac{1}{3^4}=\\cfrac{1}{81}$$\u3068\u306a\u308a\u307e\u3059\u3002(2)\u306b\u9032\u3080\u3068\u72b6\u6cc1\u306f\u5909\u308f\u308a\u307e\u3059\u3002\u4eca\u56de\u306e\u8a2d\u5b9a\u3067\u306f$1$\u4eba\u304c\u52dd\u3064\u306e\u306f\u30eb\u30fc\u30eb\u2461\u3068\u30eb\u30fc\u30eb\u2462\u306e\u72b6\u6cc1\u304c\u5b58\u5728\u3057\u307e\u3059\u3002\u30eb\u30fc\u30eb\u2461\u306e\u5834\u5408,\u52dd\u8005\u306e\u9078\u3073\u65b9\u306f$5$\u901a\u308a\u3067,\u624b\u306e\u51fa\u3057\u65b9\u306f$3$\u901a\u308a\u306a\u306e\u3067,\u5168\u90e8\u3067$5\\times3=15$\u901a\u308a\u3002<\/p>\n
\u30eb\u30fc\u30eb\u2462\u306e\u5834\u5408,\u52dd\u8005\u306e\u9078\u3073\u65b9\u306f$5$\u901a\u308a\u3067,\u624b\u306e\u51fa\u3057\u65b9\u306f$3$\u901a\u308a\u3002
\n\u4ed6\u306e$4$\u4eba\u306e\u624b\u306e\u51fa\u3057\u65b9\u306b\u3064\u3044\u3066\u8003\u3048\u308b\u3068,$1$\u4eba\u304c\u30b0\u30fc\u3067\u52dd\u8005\u3068\u306a\u308b\u5834\u5408,\u6b8b\u308a$4$\u4eba\u4e2d$2$\u4eba\u304c\u30d1\u30fc\u3067,$2$\u4eba\u304c\u30c1\u30e7\u30ad\u3092\u51fa\u3057\u3066\u3044\u307e\u3059\u3002\u30d1\u30fc\u3092\u51fa\u3059\u4eba\u306e\u9078\u3073\u65b9\u306f${}_4{\\rm C}_2=6$\u901a\u308a\u3002\u3053\u306e\u6bb5\u968e\u3067,\u6b8b\u308a$2$\u4eba\u306e\u624b\u306f\u30c1\u30e7\u30ad\u306e$1$\u901a\u308a\u306b\u306a\u308a\u307e\u3059\u3002
\n$1$\u4eba\u304c\u30d1\u30fc\u3067\u52dd\u8005\u306b\u306a\u308b\u5834\u5408\u3068\u30c1\u30e7\u30ad\u3067\u52dd\u8005\u306b\u306a\u308b\u5834\u5408\u3082\u540c\u3058\u306a\u306e\u3067$5\\times3\\times{}_4{\\rm C}_2=90$\u901a\u308a\u3002<\/p>\n
\u4ee5\u4e0a\u3088\u308a,$1$\u4eba\u304c\u52dd\u8005\u3068\u306a\u308b\u306e\u306f$15+90=105$\u901a\u308a\u3068\u306a\u308a\u307e\u3059\u3002
\n\u6c42\u3081\u308b\u78ba\u7387\u306f$\\cfrac{105}{243}=\\cfrac{35}{81}$\u3067\u3059\u3002<\/p>\n
\u3044\u3088\u3044\u3088(3)\u3067\u3059\u3002
\n\u52dd\u8005\u306e\u4eba\u6570\u306f$0$, $1$, $2$, $3$, $4$\u306e$5$\u7a2e\u985e\u3042\u308a\u307e\u3059\u304b\u3089,\u3053\u306e\u4e2d\u3067\u8a08\u7b97\u3057\u3084\u3059\u3044$4$\u7a2e\u985e\u3092\u8a08\u7b97\u3057\u307e\u3057\u3087\u3046\u3002
\n$0$\u4eba\u3068$1$\u4eba\u304c\u52dd\u8005\u3068\u306a\u308b\u78ba\u7387\u306f\u305d\u308c\u305e\u308c(1)$\\cfrac{1}{81}$,(2)$\\cfrac{35}{81}$\u3067\u3059\u3002<\/p>\n
\u52dd\u8005\u304c$3$\u4eba, $4$\u4eba\u3068\u306a\u308b\u5834\u5408\u306f\u30eb\u30fc\u30eb\u2461\u300c2\u7a2e\u985e\u306e\u624b\u304c\u51fa\u305f\u3068\u304d\u300d\u306b\u9650\u308a\u307e\u3059\u3002
\n\u4e00\u65b9\u3067\u52dd\u8005\u304c$2$\u4eba\u3068\u306a\u308b\u306e\u306f,\u30eb\u30fc\u30eb\u2461\u300c2\u7a2e\u985e\u306e\u624b\u304c\u51fa\u305f\u3068\u304d\u300d\u3068\u30eb\u30fc\u30eb\u2462\u300c3\u7a2e\u985e\u306e\u624b\u304c\u51fa\u305f\u3068\u304d\u300d\u306e\u72b6\u6cc1\u304c\u8003\u3048\u3089\u308c\u307e\u3059\u3002
\n\u3053\u306e\u5834\u5408\u306f,\u52dd\u8005\u304c$3$\u4eba, $4$\u4eba\u3068\u306a\u308b\u5834\u5408\u3092\u6c42\u3081\u308b\u306e\u304c\u901f\u305d\u3046\u3067\u3059\u3002<\/p>\n
\u52dd\u8005\u304c$3$\u4eba\u3068\u306a\u308b\u306e\u306f,\u52dd\u8005\u306e\u9078\u3073\u65b9\u304c${}_5{\\rm C}_3=10$\u901a\u308a\u3067,\u624b\u306e\u51fa\u3057\u65b9\u304c$3$\u901a\u308a\u306a\u306e\u3067$10\\times3=30$\u901a\u308a\u3002
\n\u52dd\u8005\u304c$4$\u4eba\u3068\u306a\u308b\u306e\u306f,\u52dd\u8005\u306e\u9078\u3073\u65b9\u304c${}_5{\\rm C}_4=5$\u901a\u308a\u3067,\u624b\u306e\u51fa\u3057\u65b9\u304c$3$\u901a\u308a\u306a\u306e\u3067$5\\times3=15$\u901a\u308a\u3002
\n\u305d\u308c\u305e\u308c\u78ba\u7387\u306f$\\cfrac{30}{243}=\\cfrac{10}{81}$,$\\cfrac{15}{243}=\\cfrac{5}{81}$\u3068\u306a\u308a\u307e\u3059\u3002
\n\u52dd\u8005\u304c$2$\u4eba\u3068\u306a\u308b\u78ba\u7387\u306f$$1-\\cfrac{1}{81}-\\cfrac{35}{81}-\\cfrac{10}{81}-\\cfrac{5}{81}=\\cfrac{30}{81}$$\u3068\u6c42\u3081\u3089\u308c\u307e\u3059\u3002<\/p>\n
\u4ee5\u4e0a\u3088\u308a\u52dd\u8005\u306e\u4eba\u6570\u306e\u671f\u5f85\u5024\u306f
\n$$0\\times\\cfrac{1}{81}+1\\times\\cfrac{35}{81}+2\\times\\cfrac{30}{81}+3\\times\\cfrac{10}{81}+4\\times\\cfrac{5}{81}=\\cfrac{145}{81}$$
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/details>\n\n
\u5f0f\u306e\u5f62\u306b\u6ce8\u76ee\u3057\u306a\u304c\u3089\u5909\u5f62\u3092\u8a66\u307f\u308b<\/h3>\n
\u6570\u5b66\u2162\u306e\u7bc4\u56f2\u306e\u554f\u984c\u3092\u898b\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n
\n
\u4f8b\u984c<\/div>\n
\n
(1)$x\\geqq0$\u306e\u3068\u304d, \u4e0d\u7b49\u5f0f$x-\\cfrac{1}{2}x^2\\leqq \\log(1+x)\\leqq x$\u3092\u793a\u305b\u3002
\n(2)$a_n=\\left(1+\\cfrac{1}{n^2}\\right)\\left(1+\\cfrac{2}{n^2}\\right)\\cdots\\cdots\\left(1+\\cfrac{n}{n^2}\\right)$\u3068\u3059\u308b\u3068\u304d, $\\displaystyle\\lim_{n\\to\\infty}a_n$\u3092\u6c42\u3081\u3088\u3002<\/p>\n<\/div>\n<\/div>\n
(1)\u3067\u4e0d\u7b49\u5f0f\u3092\u793a\u3057, (2)\u3068\u306e\u3064\u306a\u304c\u308a\u3092\u8003\u3048\u3066\u307f\u307e\u3059\u3002<\/p>\n
(2)\u3067\u306f$\\log$\u304c\u898b\u5f53\u305f\u308a\u307e\u305b\u3093\u3002\u5f37\u5f15\u306b$\\log$\u3092\u3064\u3051\u3066$\\log a_n$\u3092\u8003\u3048\u308b\u3053\u3068\u306b\u3088\u308a,
\n$\\begin{align*}
\n\\log a_n&=\\log\\left(1+\\cfrac{1}{n^2}\\right)+\\log\\left(1+\\cfrac{2}{n^2}\\right)\\cdots\\cdots+\\log\\left(1+\\cfrac{n}{n^2}\\right)\\\\
\n&=\\sum_{k=1}^n\\log\\left(1+\\cfrac{k}{n^2}\\right)\\end{align*}$
\n\u3068\u306a\u308a\u7a4d\u304c\u548c\u3067\u8868\u305b\u307e\u3059\u3002
\n\u3053\u306e\u3068\u304d, $(1)$\u306e\u4e0d\u7b49\u5f0f\u306b\u304a\u3044\u3066$x=\\cfrac{k}{n^2}$\u3092\u304a\u3044\u3066\u307f\u3088\u3046\u3068\u6c7a\u307e\u308a\u307e\u3059\u3002
\n\u5b9f\u969b,
\n$$\\cfrac{k}{n^2}-\\cfrac{1}{2}\\left(\\cfrac{k}{n^2}\\right)^2\\leqq \\log\\left(1+\\cfrac{k}{n^2}\\right)\\leqq \\cfrac{k}{n^2}$$
\n\u3068\u306a\u308a,$k$=$1$,$2$,$\\cdots\\cdots$,$n$\u3068\u3057\u3066\u52a0\u3048\u308c\u3070
\n$$\\sum_{k=1}^n\\left\\{\\cfrac{k}{n^2}-\\cfrac{1}{2}\\left(\\cfrac{k}{n^2}\\right)^2\\right\\}\\leqq\\sum_{k=1}^n\\log\\left(1+\\cfrac{k}{n^2}\\right)\\leqq\\sum_{k=1}^n\\cfrac{k}{n^2}$$
\n\u3057\u305f\u304c\u3063\u3066
\n$$\\sum_{k=1}^n\\left\\{\\cfrac{k}{n^2}-\\cfrac{1}{2}\\left(\\cfrac{k}{n^2}\\right)^2\\right\\}\\leqq\\log a_n\\leqq\\sum_{k=1}^n\\cfrac{k}{n^2}$$
\n\u304c\u5f97\u3089\u308c\u307e\u3059\u3002
\n$n\\to\\infty$\u3068\u3059\u308b\u3068, \u4e0d\u7b49\u5f0f\u306e\u6700\u5de6\u8fba\u3068\u6700\u53f3\u8fba\u306f<\/p>\n
$\\begin{align*}
\n&\\displaystyle\\sum_{k=1}^n\\left\\{\\cfrac{k}{n^2}-\\cfrac{1}{2}\\left(\\cfrac{k}{n^2}\\right)^2\\right\\}\\\\
\n=&\\cfrac{1}{n^2}\\cdot\\cfrac{1}{2}n(n+1)-\\cfrac{1}{2}\\cdot\\cfrac{1}{n^4}\\cdot\\cfrac{1}{6}n(n+1)(2n+1)\\to\\cfrac{1}{2}\\end{align*}$<\/p>\n
$\\displaystyle\\sum_{k=1}^n\\cfrac{k}{n^2}=\\cfrac{1}{n^2}\\cdot\\cfrac{1}{2}n(n+1)\\to\\cfrac{1}{2}$<\/p>\n
\u3068\u306a\u308b\u304b\u3089, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a
\n$$\\lim_{n\\to\\infty}\\log a_n=\\cfrac{1}{2}$$
\n\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3088\u3063\u3066,
\n$$\\lim_{n\\to\\infty}a_n=\\sqrt{e}$$
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n
\n
\n
\u3053\u306e\u3088\u3046\u306b, \u793a\u3057\u305f\u3082\u306e, \u6c42\u3081\u305f\u3082\u306e\u304c\u4f7f\u3048\u308b\u3088\u3046\u306b\u5909\u5f62\u3092\u8003\u3048\u308b\u3053\u3068\u304c\u8981\u6c42\u3055\u308c\u307e\u3059\u3002<\/div>\n<\/div>\n<\/div>\n\n
\u554f\u984c\u306b\u884c\u304d\u8a70\u3063\u305f\u3089\u524d\u306e\u554f\u984c\u3092\u78ba\u8a8d\u3059\u308b<\/h3>\n
\u554f\u984c\u3092\u89e3\u304d\u9032\u3081\u308b\u3068\u304d\u306b,\u524d\u306e\u554f\u984c\u3068\u306e\u95a2\u9023\u304c\u8aad\u307f\u53d6\u308c\u306a\u3044\u3068\u304d\u306f\u7d20\u76f4\u306b\u554f\u984c\u3092\u89e3\u304d\u9032\u3081\u307e\u3057\u3087\u3046\u3002\u305d\u306e\u4e0a\u3067,\u65b9\u91dd\u304c\u7acb\u305f\u306a\u304b\u3063\u305f\u308a,\u89e3\u304d\u9032\u3081\u3089\u308c\u306a\u304b\u3063\u305f\u308a\u3059\u308b\u5834\u5408\u306f\u524d\u306e\u554f\u984c\u3092\u6539\u3081\u3066\u898b\u3066\u307f\u307e\u3059\u3002<\/p>\n
\n
\n
\u554f\u984c\u3092\u89e3\u304d\u9032\u3081\u305f\u3053\u3068\u3067\u6700\u521d\u306e\u5f0f\u3068\u306f\u9055\u3046\u5f0f,\u72b6\u6cc1\u306b\u306a\u3063\u3066\u3044\u3066,\u524d\u306e\u554f\u984c\u3068\u306e\u95a2\u9023\u304c\u898b\u3048\u3066\u304f\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u3002<\/div>\n<\/div>\n<\/div>\n\n
\u304a\u308f\u308a\u306b<\/h2>\n
\u79cb\u306b\u306a\u308a,\u305d\u308d\u305d\u308d\u904e\u53bb\u554f\u306b\u89e6\u308c\u308b\u6a5f\u4f1a\u304c\u5897\u3048\u3066\u304f\u308b\u3068\u601d\u3044\u307e\u3059\u3002\u5b9f\u969b\u306e\u5165\u8a66\u554f\u984c\u306b\u89e6\u308c,\u4eca\u56de\u304a\u8a71\u3057\u305f\u300c\u8a98\u5c0e\u306b\u4e57\u308b\u300d\u3068\u3044\u3046\u3053\u3068\u304c\u3069\u3046\u3044\u3046\u3053\u3068\u306a\u306e\u304b\u4f53\u611f\u3057\u3066\u3082\u3089\u3044\u305f\u3044\u3067\u3059\u3002<\/p>\n
\u8a98\u5c0e\u306e\u306a\u3044\u554f\u984c\u306f,\u3044\u308d\u3044\u308d\u306a\u3053\u3068\u3092\u72ec\u81ea\u306b\u601d\u3044\u4ed8\u304d,\u8ad6\u8a3c\u3057\u305f\u308a\u8a08\u7b97\u3057\u3066\u6c42\u3081\u308b\u3053\u3068\u306b\u306a\u308b\u306e\u3067\u96e3\u6613\u5ea6\u304c\u9ad8\u3044\u3067\u3059\u3002\u4e00\u3064\u306e\u7d50\u8ad6\u3092\u5f97\u308b\u305f\u3081\u306b,\u524d\u306b\u3044\u304f\u3064\u304b\u554f\u984c\u304c\u3064\u3044\u3066\u3044\u308b\u3068\u304d\u306f\u300c\u8a98\u5c0e\u304b\u306a\uff1f\u300d\u3068\u7591\u3063\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n
\u3053\u308c\u3092\u6a5f\u306b,\u554f\u984c\u306e\u69cb\u6210\u306b\u6ce8\u76ee\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n
\u305d\u308c\u3067\u306f, \u4eca\u56de\u306f\u3053\u306e\u8fba\u3067\uff01<\/p>\n
![\"\"](\"https:\/\/www.educational-lounge.com\/wp-content\/uploads\/2023\/01\/b2a07ce9e24a798a7886e5d0e73ded79_s.jpg\")
![\"\u30b0\u30e9\u30d5\u3092\u5229\u7528\u3057\u3066\u8003\u3048\u308b\uff01\u3000\u5165\u8a66\u983b\u51fa\u30c6\u30fc\u30de\u300c\u4e0d\u7b49\u5f0f\u306e\u6210\u7acb\u6761\u4ef6\u300d\u306e\u89e3\u6cd5\"](\"https:\/\/www.educational-lounge.com\/wp-content\/uploads\/2024\/04\/eyecatch_math_2404-150x150.jpg\")