{"id":14944,"date":"2010-11-16T19:54:42","date_gmt":"2010-11-16T11:54:42","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=14944"},"modified":"2021-10-06T16:30:44","modified_gmt":"2021-10-06T08:30:44","slug":"%e3%80%8c%e6%95%b8%e5%ad%b8%e6%ad%b8%e7%b4%8d%e6%b3%95%e3%80%8d%e5%85%b6%e5%af%a6%e4%b8%8d%e6%ad%b8%e7%b4%8d%ef%bc%88mathematical-induction-is-not-induction-at-all%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%e3%80%8c%e6%95%b8%e5%ad%b8%e6%ad%b8%e7%b4%8d%e6%b3%95%e3%80%8d%e5%85%b6%e5%af%a6%e4%b8%8d%e6%ad%b8%e7%b4%8d%ef%bc%88mathematical-induction-is-not-induction-at-all%ef%bc%89\/","title":{"rendered":"\u300c\u6578\u5b78\u6b78\u7d0d\u6cd5\u300d\u5176\u5be6\u4e0d\u6b78\u7d0d"},"content":{"rendered":"

\u300c\u6578\u5b78\u6b78\u7d0d\u6cd5\u300d\u5176\u5be6\u4e0d\u6b78\u7d0d\uff08Mathematical Induction is not induction at all\uff09<\/strong><\/span>
\n\u4e2d\u592e\u7814\u7a76\u9662\u6578\u5b78\u6240\u674e\u570b\u5049\u7814\u7a76\u54e1\/\u4e2d\u592e\u7814\u7a76\u9662\u6578\u5b78\u6240\u674e\u570b\u5049\u7814\u7a76\u54e1\u8cac\u4efb\u7de8\u8f2f<\/span><\/strong><\/span><\/p>\n

\u901a\u5e38\u6559\u79d1\u66f8\u88e1\u8b1b\u89e3\u6578\u5b78\u6b78\u7d0d\u6cd5\u6642\uff0c\u6703\u51fa\u73fe\u985e\u4f3c\u4e0b\u9762\u7684\u9673\u8ff0\uff1a<\/span><\/span>\u7d66\u5b9a\u4e00\u500b\u95dc\u65bc\u6b63\u6574\u6578 $${n}$$ \u7684\u6558\u8ff0 $${P(n)}$$\uff0c\u5982\u679c\u60f3\u8b49\u660e $${P(n)}$$ \u5c0d\u65bc\u6240\u6709\u6b63\u6574\u6578 $${n}$$ \u90fd\u6210\u7acb\uff0c\u53ea\u9700\u505a\u5169\u4ef6\u4e8b\uff1a<\/p>\n

\uff11\u3001\uff08\u6b78\u7d0d\u57fa\u790e\uff09\u8b49\u660e $${n}=1$$ \u6642\uff0c$${P(1)}$$ \u6210\u7acb\u3002<\/span><\/span>
\n\uff12\u3001\uff08\u6b78\u7d0d\u6b65\u9a5f\uff09\u5047\u8a2d $${n}={k}$$ \u6642 $${P(k)}$$\u6210\u7acb\uff0c\u8b49\u660e\u7576 $${n=k+1}$$ \u6642\uff0c$${P(k+1)}$$ \u6210\u7acb\u3002<\/span><\/span><\/p>\n

\u7576\u9019\u5169\u4ef6\u4efb\u52d9\u5b8c\u6210\u5f8c\uff0c$${P(n)}$$ \u5c31\u5c0d\u6240\u6709\u6b63\u6574\u6578 $${n}$$ \u6210\u7acb\u4e86\u3002<\/span><\/span><\/p>\n

\u5728\u8b49\u660e\u6d89\u53ca\u6b63\u6574\u6578\u7684\u547d\u984c\u6642\uff0c\u6578\u5b78\u6b78\u7d0d\u6cd5\u662f\u975e\u5e38\u6709\u5a01\u529b\u7684\u5de5\u5177\u3002\u53ef\u662f\u5b78\u751f\u521d\u6b21\u63a5\u89f8\u9019\u7a2e\u8ad6\u8b49\u7684\u65b9\u5f0f\u6642\uff0c\u5f80\u5f80\u6703\u7522\u751f\u4e00\u4e9b\u56f0\u60d1\u3002\u5b78\u751f\u770b\u5b8c\u9019\u7a2e\u9673\u8ff0\uff0c\u807d\u5b8c\u8001\u5e2b\u7684\u8b1b\u89e3\uff0c\u4ee5\u53ca\u4f5c\u5b8c\u4e00\u4e9b\u4f8b\u984c\u4e4b\u5f8c\uff0c\u53ef\u80fd\u4ecd\u611f\u89ba\u5230\u628a $${n}$$ \u5beb\u6210 $${k}$$ \u597d\u4f3c\u8b8a\u9b54\u8853\u3002\u7576 $${n}={k}$$ \u6642\uff0c$${P(k)}$$ \u4e0d\u5c31\u662f $${P(n)}$$ \u4e86\u55ce\uff1f\u90a3\u4f60\u5df2\u7d93\u7528\u4e86 $${P(n)}$$ \u6210\u7acb\u4e86\uff0c\u5e79\u561b\u9084\u8981\u518d\u8b49\u660e $${P(n)}$$ \u90fd\u6210\u7acb\u5462\uff1f<\/span><\/span><\/p>\n

\u89e3\u9664\u9019\u500b\u56f0\u60d1\u7684\u95dc\u9375\u5728\u65bc\u7406\u89e3\u300c\u6b78\u7d0d\u6b65\u9a5f\u300d\u662f\u4e00\u500b\u4f7f\u7528\u300c\u771f\u503c\u860a\u6db5\u300d\u7684\u689d\u4ef6\u53e5\uff0c\u5982\u679c\u9019\u4e00\u6b65\u4f5c\u5f97\u6b63\u78ba\uff0c\u4e26\u4e0d\u4fdd\u8b49\u689d\u4ef6\u53e5\u7684\u524d\u63d0\u6c38\u9060\u70ba\u771f\u3002\u53ea\u4fdd\u8b49\u7576\u524d\u63d0\u70ba\u771f\u6642\uff0c\u689d\u4ef6\u53e5\u7684\u7d50\u8ad6\u4e5f\u70ba\u771f\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u5373\u4f7f $${P(n)}\\rightarrow{P(n+1)}$$ \u5c0d\u65bc\u6240\u6709 $${n}$$ \u5747\u70ba\u771f\uff0c\u4e26\u4e0d\u5fc5\u7136\u63a8\u5c0e\u51fa $${P(n)}$$ \u5c0d\u65bc\u6240\u6709 $${n}$$ \u70ba\u771f\u3002<\/span><\/span><\/p>\n

\u6211\u5011\u4f86\u770b\u5169\u500b\u6709\u6bdb\u75c5\u7684\u4f8b\u5b50\uff1a<\/p>\n

\u932f\u4f8b\uff11\uff1a\u8a66\u8b49 $$\\displaystyle 1+2+3\\mbox{…}+n=\\frac{n(n+1)}{2}+1$$
\n\u932f\u8b49\uff1a<\/p>\n

\u4ee4 $${P(n)}$$ \u8868\u793a\u6558\u8ff0 $$\\displaystyle 1+2+3\\mbox{…}+n=\\frac{n(n+1)}{2}+1$$\u3002<\/p>\n

\u5047\u8a2d $${n}={k}$$ \u6642 $${P(k)}$$ \u6210\u7acb\uff0c\u5373\u00a0$$\\displaystyle 1+2+3\\mbox{…}+k=\\frac{k(k+1)}{2}+1$$\u3002<\/p>\n

\u7576 $${n}={k+1}$$ \u6642\uff0c$$\\displaystyle 1+2+3\\mbox{…}+k+(k+1)=\\frac{k(k+1)}{2}+1+(k+1)=\\frac{(k+1)(k+2)}{2}+1$$<\/p>\n

\u5373 $${P(k+1)}$$ \u6210\u7acb\uff0c\u6240\u4ee5 $${P(n)}$$ \u5c0d\u65bc\u6240\u6709\u6b63\u6578 $${n}$$ \u90fd\u6210\u7acb\u3002\uff08\u8b49\u7562\uff09<\/p>\n

\u9019\u500b\u4f8b\u5b50\u901a\u5e38\u662f\u7528\u4f86\u8aaa\u660e\u6c92\u6709\u9a57\u8b49\u300c\u6b78\u7d0d\u57fa\u790e\u300d\u7684\u932f\u8aa4\uff0c\u56e0\u70ba $${P(1)}$$ \u5176\u5be6\u65b7\u8a00\u4e00\u500b\u8352\u8b2c\u7684\u7d50\u8ad6 $${1=\\frac{1(1+1)}{2}+1=2}$$\u3002\u800c\u6211\u5011\u5728\u6b64\u8981\u5f37\u8abf\u7684\u662f\uff0c\u932f\u8b49\u96d6\u7136\u6c92\u6709\u6b63\u78ba\u4f7f\u7528\u6578\u5b78\u6b78\u7d0d\u6cd5\uff0c\u53ef\u662f\u5f9e $${P(k)}$$ \u5230 $${P(k+1)}$$ \u7684\u63a8\u8ad6\u672c\u8eab\u4e26\u6c92\u6709\u932f\u8aa4\u3002\u6839\u64da\u300c\u771f\u503c\u860a\u6db5\u300d\u7684\u898f\u5b9a\uff0c\u689d\u4ef6\u53e5\u7684\u524d\u63d0\u8207\u7d50\u8ad6\u90fd\u70ba\u5047\u6642\uff0c\u689d\u4ef6\u53e5\u672c\u8eab\u9084\u662f\u70ba\u771f\u3002<\/p>\n

\u932f\u4f8b\uff12\uff1a\u8a66\u8b49\u300c\u4efb\u4f55 $${n}$$ \u500b\u4eba\u90fd\u540c\u9ad8\u300d\u3002
\n\u932f\u8b49\uff1a<\/p>\n

\u4ee4 $${P(n)}$$ \u8868\u793a\u6558\u8ff0\u300c\u4efb\u4f55 $${n}$$ \u500b\u4eba\u90fd\u540c\u9ad8\u300d\u3002<\/p>\n

\u7576 $${n=1}$$\u6642\uff0c\u6558\u8ff0\u8b8a\u70ba\u300c\u4efb\u4f55\u4e00\u500b\u4eba\u90fd\u540c\u9ad8\u300d\uff0c\u986f\u7136\u6210\u7acb\u3002<\/p>\n

\u5047\u8a2d $${n=k}$$ \u6642\u6558\u8ff0\u6210\u7acb\uff0c\u5373\u300c\u4efb\u4f55 $${k}$$ \u500b\u4eba\u90fd\u540c\u9ad8\u300d\uff0c\u90a3\u9ebc\u7576 $${n=k+1}$$ \u6642\uff0c<\/p>\n

\u5c07 $${k+1}$$ \u500b\u4eba\u8a18\u70ba $$A_1,A_2,\\mbox{…},A_k,A_{k+1}$$\uff0c\u7531\u6b78\u7d0d\u6cd5\u5047\u8a2d\u53ef\u77e5 $$A_1,A_2,\\mbox{…},A_k$$ \u90fd\u540c\u9ad8\uff0c<\/p>\n

\u800c $$A_2,\\mbox{…},A_k,A_{k+1}$$ \u4e5f\u90fd\u540c\u9ad8\uff0c\u6240\u4ee5 $$A_1,A_2,\\mbox{…},A_k,A_{k+1}$$ \u90fd\u540c\u9ad8\u3002<\/p>\n

\u7d93\u7531\u4ee5\u4e0a\u7684\u300c\u6b78\u7d0d\u57fa\u790e\u300d\u8207\u300c\u6b78\u7d0d\u6b65\u9a5f\u300d\u5169\u968e\u6bb5\uff0c$${P(n)}$$\u5c0d\u65bc\u6240\u6709\u6b63\u6574\u6578$${n}$$\u90fd\u6210\u7acb\u3002\uff08\u9a57\u7562\uff09<\/p>\n

\u9019\u500b\u4f8b\u5b50\u5f62\u5f0f\u4e0a\u5177\u6709\u6578\u5b78\u6b78\u7d0d\u6cd5\u7684\u5169\u9805\u8981\u4ef6\u300c\u6b78\u7d0d\u57fa\u790e\u300d\u548c\u300c\u6b78\u7d0d\u6b65\u9a5f\u300d\uff0c\u53ef\u662f\u5728\u300c\u6b78\u7d0d\u6b65\u9a5f\u300d\u6642\u6240\u4f5c\u7684\u63a8\u8ad6\u662f\u6709\u554f\u984c\u7684\u3002\u8981\u60f3\u4f7f\u63a8\u8ad6\u6b63\u78ba\uff0c\u5fc5\u9808 $${k}\\ge 3$$\u3002\u5982\u6b64\u4e0d\u7ba1\u62ff\u6389\u5c3e\u5df4\u90a3\u500b\u4eba\uff0c\u6216\u662f\u62ff\u6389\u9818\u982d\u90a3\u500b\u4eba\uff0c\u90fd\u9084\u6709\u4eba\u5269\u4e0b\u7576\u4f5c\u5171\u540c\u6bd4\u8f03\u7684\u4f9d\u64da\u3002\u9019\u500b\u4f8b\u5b50\u88e1\u5f9e $${P(k)}$$ \u5230 $${P(k+1)}$$ \u7684\u63a8\u8ad6\u672c\u8eab\u5c31\u4e0d\u70ba\u771f\uff0c\u81ea\u7136\u4e5f\u7121\u6cd5\u4fdd\u8b49 $${P(n)}$$ \u5c0d\u65bc\u6240\u6709\u6b63\u6574\u6578 $${n}$$ \u90fd\u6210\u7acb\u4e86\u3002<\/p>\n

\u6709\u6559\u79d1\u66f8\u5c0d\u65bc\u300c\u6b78\u7d0d\u6cd5\u300d\u7684\u610f\u7fa9\u4f5c\u51fa\u5982\u4e0b\u7684\u8aaa\u660e\uff1a\u300c\u6b78\u7d0d\u6cd5\u662f\u5c0d\u4e00\u4e9b\u500b\u5225\u7684\u4e8b\u7269\u8a73\u52a0\u89c0\u5bdf\u3001\u7d00\u9304\uff0c\u63a2\u8a0e\u9019\u4e9b\u500b\u5225\u4e8b\u7269\u7684\u5171\u540c\u7279\u6027\uff0c\u7136\u5019\u5c07\u6240\u5f97\u5230\u7d50\u679c\u63a8\u5ee3\u5230\u5176\u4ed6\u672a\u7d93\u89c0\u5bdf\u7684\u540c\u985e\u4e8b\u7269\u4e2d\uff0c\u5f9e\u4e2d\u6982\u62ec\u51fa\u540c\u985e\u4e8b\u7269\u4e00\u9805\u666e\u904d\u6027\u7d50\u8ad6\u3002\u300d<\/p>\n

\u9019\u662f\u5c0d\u65bc\u4e00\u822c\u79d1\u5b78\u65b9\u6cd5\u88e1\u7684\u300c\u6b78\u7d0d\u6cd5\u300d\u7684\u8aaa\u660e\uff0c\u5176\u5be6\u300c\u6578\u5b78\u6b78\u7d0d\u6cd5\u300d\u4e26\u6c92\u6709\u57f7\u884c\u4efb\u4f55\u6b78\u7d0d\uff0c\u5b83\u7d14\u7cb9\u662f\u689d\u4ef6\u53e5\u7684\u904b\u7528\uff0c\u662f\u5c6c\u65bc\u6f14\u7e79\u908f\u8f2f\u7684\u7bc4\u570d\u3002\u4ee5\u300c\u6578\u5b78\u6b78\u7d0d\u6cd5\u300d\u8b49\u660e\u7684\u547d\u984c\uff0c\u4e5f\u8a31\u66fe\u7d93\u904b\u7528\u300c\u6b78\u7d0d\u6cd5\u300d\u5e6b\u5fd9\u627e\u51fa\u6070\u7576\u7684\u6558\u8ff0\u5f62\u5f0f\uff0c\u4f46\u662f\u8b49\u660e\u7684\u904e\u7a0b\u537b\u662f\u6f14\u7e79\uff0c\u6f14\u7e79\uff0c\u518d\u6f14\u7e79\u3002<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u8aaa\u660e\u6578\u5b78\u6b78\u7d0d\u6cd5\u7684\u8b49\u660e\u904e\u7a0b\u4e3b\u8981\u662f\u689d\u4ef6\u53e5\u7684\u904b\u7528\uff0c\u5c6c\u65bc\u6f14\u7e79\u6cd5\uff0c\u800c\u975e\u4e00\u822c\u4eba\u8a8d\u70ba\u7684\u6b78\u7d0d\u6cd5\u3002 <\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111,216,215],"tags":[418,344,420,419],"class_list":["post-14944","post","type-post","status-publish","format-standard","hentry","category-mathematics00","category-math02","category-math01","tag-418","tag-344","tag-420","tag-419","loop-entry","cat-111","cat-216","cat-215","no-thumbnail"],"views":9061,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/14944","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=14944"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/14944\/revisions"}],"predecessor-version":[{"id":89341,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/14944\/revisions\/89341"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=14944"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=14944"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=14944"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}