{"id":2852,"date":"2025-08-23T01:42:51","date_gmt":"2025-08-22T16:42:51","guid":{"rendered":"https:\/\/math-friend.com\/?p=2852"},"modified":"2025-08-23T01:44:27","modified_gmt":"2025-08-22T16:44:27","slug":"%e3%80%90%e6%9d%b1%e5%8c%97%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e6%8c%87%e6%95%b0%e3%81%a8%e5%a4%9a%e9%a0%85%e5%bc%8f%e3%81%ae%e5%a4%a7%e5%b0%8f%e6%af%94%e8%bc%83%e3%81%a8%e6%95%b4%e6%95%b0","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=2852","title":{"rendered":"\u3010\u6771\u5317\u5927\u5b66\u5165\u8a66\u3011\u6307\u6570\u3068\u591a\u9805\u5f0f\u306e\u5927\u5c0f\u6bd4\u8f03\u3068\u6574\u6570\u306e\u95a2\u4fc2\u5f0f(2020)"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\\(n\\)\u3092\u6b63\u306e\u6574\u6570, \\(a\\), \\(b\\)\u3092\\(0\\)\u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.
(1) \\(n\\geq 3\\)\u306e\u3068\u304d, \\(2^n+n^2+8<3^n\\)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.
(2) \\(2^n+n^2+8\\geq 3^n\\)\u3092\u6e80\u305f\u3059\\(n\\)\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.
(3) \\(2^n+n^2+8= 3^n+an+b\\)\u3092\u6e80\u305f\u3059\\(a\\), \\(b\\), \\(n\\)\u306e\u7d44\\((a,b,n)\\)\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.
(2020 \u6771\u5317\u5927\u5b66 \u6587\u7cfb [2])<\/span><\/p>\n\n\n\n

\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

(1) \\(f(n)=3^n-(2^n+n^2+8)\\)\u3068\u304a\u304d, \\(n\\geq 3\\)\u306e\u3068\u304d\\(f(n)>0\\)\u3067\u3042\u308b\u3053\u3068\u3092, \u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u308a\u8a3c\u660e\u3059\u308b.
\u2460 \\(n=3\\)\u306e\u3068\u304d$$
f(3)=3^3-(2^3+3^2+8)=2>0
$$\u3068\u306a\u3063\u3066, \\(n=3\\)\u306e\u3068\u304d\\(f(n)>0\\)\u304c\u6210\u308a\u7acb\u3064.

\u2461 \\(n=k\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057, \\(n=k+1\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3059
\u4eee\u5b9a\u304b\u3089, \\(f(k)=3^k-(2^k+k^2+8)>0\\)\u3088\u308a, \\(3^k>2^k+k^2+8\\)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066,$$
\\begin{align}
f(k+1)&=3^{k+1}-\\left\\{2^{k+1}+(k+1)^2+8\\right\\}\\\\[1.5ex]
&=3\\cdot 3^k-\\left\\{2^{k+1}+(k+1)^2+8\\right\\}\\\\[1.5ex]
&>3(2^k+k^2+8)-\\left\\{2^{k+1}+(k+1)^2+8\\right\\}\\\\[1.5ex]
&=2^k+2k^2-2k+15\\\\[1.5ex]
&=2^k+k^2+(k-1)^2+14>0
\\end{align}
$$\u3068\u306a\u308a, \\(n=k+1\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064.

\u2460, \u2461 \u3088\u308a\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u304b\u3089\\(n\\geq 3\\)\u3092\u6e80\u305f\u3059\\(n\\)\u306b\u5bfe\u3057\u3066, \\(f(n)>0\\), \u3064\u307e\u308a, \\(2^n+n^2+8<3^n\\)\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u3055\u308c\u305f.<\/p>\n\n\n\n

(2) (1)\u304b\u3089\\(n\\geq 3\\)\u306e\u3068\u304d, \\(2^n+n^2+8<3^n\\)\u304c\u6210\u308a\u7acb\u3064\u304b\u3089, \\(2^n+n^2+8\\geq 3^n\\)\u3068\u306a\u308b\\(n\\)\u306e\u5019\u88dc\u306f\\(n=1\\), \\(n=2\\)\u306e\u307f\u3067\u3042\u308b.

\u30fb\\(n=1\\)\u306e\u3068\u304d$$
\\begin{align}
(\u5de6\u8fba)&=2^1+1^2+8=11\\\\[1.5ex]
(\u53f3\u8fba)&=3^1=3
\\end{align}
$$\u3088\u308a\u6210\u308a\u7acb\u3064.

\u30fb\\(n=2\\)\u306e\u3068\u304d$$
\\begin{align}
(\u5de6\u8fba)&=2^2+2^2+8=16\\\\[1.5ex]
(\u53f3\u8fba)&=3^2=9
\\end{align}
$$\u3088\u308a\u6210\u308a\u7acb\u3064.

\u3088\u3063\u3066, \\(n=1,2\\).<\/p>\n\n\n\n

(3) \\(n>0\\), \\(a,b\\geq 0\\)\u3088\u308a, \\(an+b\\geq 0\\)\u3067\u3042\u308b, \u3088\u3063\u3066, \u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3068\u304d,$$
an+b=2^n+n^2+8-3^n\\geq 0
$$\u3088\u308a, $$
2^n+n^2+8\\geq 3^n
$$\u3068\u306a\u308a, (2)\u304b\u3089\u3053\u308c\u3092\u6e80\u305f\u3059\\(n\\)\u306f\\(n=1\\), \\(n=2\\)\u306b\u9650\u3089\u308c\u308b.

\u30fb\\(n=1\\)\u306e\u3068\u304d
\u7b49\u5f0f\u306f,$$
2^1+1^2+8=3^1+a\\cdot 1+b\\,\\iff\\,a+b=8
$$\u3068\u306a\u308b\u304b\u3089, \\(a\\), \\(b\\)\u306f\\(0\\)\u4ee5\u4e0a\u306e\u6574\u6570\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066, \u3053\u308c\u3092\u6e80\u305f\u3059\\((a,b)\\)\u306f,$$
\\begin{align}
(a,b)=&(0,8), (1,7), (2,6), (3,5),\\\\[1.5ex]
&(4,4), (5,3), (6,2), (7,1), (8,0)
\\end{align}
$$\u306e9\u7d44\u3067\u3042\u308b.

\u30fb\\(n=2\\)\u306e\u3068\u304d
\u7b49\u5f0f\u306f,$$
2^2+2^2+8=3^2+a\\cdot 2+b\\,\\iff\\,2a+b=7
$$\u3068\u306a\u308b\u304b\u3089, \\(a\\), \\(b\\)\u306f\\(0\\)\u4ee5\u4e0a\u306e\u6574\u6570\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066, \u3053\u308c\u3092\u6e80\u305f\u3059\\((a,b)\\)\u306f,$$
(a,b)=(0,7), (1,5), (2,3), (3,1)
$$\u306e4\u7d44\u3067\u3042\u308b.

\u4ee5\u4e0a\u304b\u3089, \u7b49\u5f0f\u3092\u6e80\u305f\u3059\\((aa,b,n)\\)\u306f,$$
\\begin{align}
(a,b,n)=&(0,8,1), (1,7,1), (2,6,1), (3,5,1), (4,4,1), \\\\[1.5ex]
&(5,3,1), (6,2,1), (7,1,1), (8,0,1), \\\\[1.5ex]
&(0,7,2), (1,5,2), (2,3,2), (3,1,2)
\\end{align}
$$\u306e13\u901a\u308a\u3067\u3042\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n