{"id":55540,"date":"2014-08-20T06:45:43","date_gmt":"2014-08-19T22:45:43","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=55540"},"modified":"2021-10-06T16:15:21","modified_gmt":"2021-10-06T08:15:21","slug":"%e4%ba%8c%e9%a0%85%e5%bc%8f%e5%ae%9a%e7%90%86%e7%9a%84%e6%8e%a8%e5%bb%a3%ef%bc%88%e4%b8%89%ef%bc%89%ef%bc%9a%e5%92%8c%e7%ae%97%e5%ae%b6%e7%9a%84%e6%95%b8%e5%ad%b8%e8%a1%a8%ef%bc%88%e4%b8%8a%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%e4%ba%8c%e9%a0%85%e5%bc%8f%e5%ae%9a%e7%90%86%e7%9a%84%e6%8e%a8%e5%bb%a3%ef%bc%88%e4%b8%89%ef%bc%89%ef%bc%9a%e5%92%8c%e7%ae%97%e5%ae%b6%e7%9a%84%e6%95%b8%e5%ad%b8%e8%a1%a8%ef%bc%88%e4%b8%8a%ef%bc%89\/","title":{"rendered":"\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u63a8\u5ee3\uff08\u4e09\uff09: \u548c\u7b97\u5bb6\u7684\u6578\u5b78\u8868\uff08\u4e0a\uff09"},"content":{"rendered":"

\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u63a8\u5ee3\uff08\u4e09\uff09\uff1a\u548c\u7b97\u5bb6\u7684\u6578\u5b78\u8868\uff08\u4e0a\uff09
\n(The generalization of Binomial theorem\uff08III\uff09\uff1athe mathematical table of wasan mathematicians)<\/strong><\/span>
\n\u81fa\u5317\u5e02\u7acb\u548c\u5e73\u9ad8\u4e2d\u9ec3\u4fca\u744b\u6559\u5e2b<\/strong><\/span><\/p>\n

\u9023\u7d50\uff1a\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u63a8\u5ee3\uff08\u4e8c\uff09\uff1a\u6709\u7406\u6578\u51aa\u6b21<\/a><\/p>\n

\u5728\u3008\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u63a8\u5ee3\uff08\u4e00\uff09<\/a>\u3009\u8207\u3008\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u63a8\u5ee3\uff08\u4e8c\uff09<\/a>\u3009\u5169\u7bc7\u6587\u7ae0\u4e2d\uff0c\u63d0\u5230\u4e86\u6c5f\u6236\u6642\u671f\u65e5\u672c\u6578\u5b78\u5bb6\uff08\u548c\u7b97\u5bb6\uff09\u5c0d\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u63a8\u5ee3\uff0c\u5305\u542b\u5229\u7528\u7121\u7aae\u7b49\u6bd4\u7d1a\u6578\u516c\u5f0f\u4ee5\u53ca\u76f4\u89c0\u5730\u4f7f\u7528\u4e86\u300c\u7121\u7aae\u591a\u9805\u5f0f\u300d\u7684\u4e58\u6cd5\uff0c\u5c07\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u5e42\u6b21\u63a8\u5ee3\u81f3\u8ca0\u6574\u6578\u7684\u60c5\u6cc1\u3002\u4e26\u4e5f\u8aaa\u660e\u4ed6\u5011\u5982\u4f55\u5229\u7528\u958b\u65b9\u6cd5\uff08\u7d9c\u5408\u9664\u6cd5\uff0c\u4ea6\u5373\u4e2d\u570b\u50b3\u5165\u7684\u8cc8\u61b2\uff0d\u970d\u7d0d\u6cd5\uff09\u5c07\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u5e42\u6b21\u63a8\u5ee3\u81f3 $$1\/2$$ \u4ee5\u53ca $$1\/n$$\u00a0\u4efb\u610f\u7684\u60c5\u6cc1\u3002<\/p>\n

\u6709\u8da3\u7684\u662f\uff0c\u6c5f\u6236\u6642\u671f\u65e5\u672c\u6578\u5b78\u5bb6\u9032\u4e00\u6b65\u767c\u5c55\u51fa\u5404\u985e\u6578\u5b78\u300c\u8868\u300d\uff0c\u7528\u4f86\u5e6b\u52a9\u8a08\u7b97\u8207\u63a8\u5ee3\u4e8c\u9805\u5f0f\u5b9a\u7406\uff0c\u4e00\u822c\u4e5f\u4f5c\u70ba\u8a18\u8f09\u6578\u5b78\u77e5\u8b58\u4e4b\u7528\u3002\u5982\u8868\u4e00\u6240\u793a\uff0c\u70ba $$(1-x)^{-k}$$\u00a0\u985e\u4e8c\u9805\u5c55\u958b\u5f0f\u4e4b\u4fc2\u6578\u8868\uff08\u9019\u88e1\u70ba\u65b9\u4fbf\u8b80\u8005\u95b1\u8b80\uff0c\u7b46\u8005\u5c07\u539f\u8868\u683c\u5167\u5bb9\u6539\u4ee5\u73fe\u4ee3\u7b26\u865f\u4f86\u8868\u793a\uff0c\u4e26\u53d7\u7bc7\u5e45\u6240\u9650\u53ea\u5217\u51fa\u7576\u4e2d\u7684\u4e00\u90e8\u4efd\uff09\uff0c\u82e5\u6211\u5011\u50c5\u770b\u6578\u5b57\u90e8\u4efd\uff0c\u5247\u7b2c\u4e00\u5217\u7684\u6578\u5b57\u70ba\u00a0$$(1-x)^{-1}$$ \u7684\u5404\u9805\u4fc2\u6578\uff1b\u7b2c\u4e8c\u5217\u70ba$$(1-x)^{-2}$$ \u7684\u5404\u9805\u4fc2\u6578\uff1b$$\\cdots$$\uff1b\u7b2c $$k$$\u00a0\u5217\u70ba\u00a0$$(1-x)^{-k}$$ \u7684\u5404\u9805\u4fc2\u6578\uff08\u7136\u8868\u4e2d\u7686\u50c5\u5217\u5230\u524d\u4e03\u9805\uff09\u3002<\/p>\n

\u6709\u4e86\u7b2c\u4e00\u5217\u4e4b\u5f8c\uff0c\u4fbf\u53ef\u4ee5\u4efb\u610f\u5730\u64f4\u5f35\u6574\u500b\u8868\u7684\u5167\u5bb9\uff0c\u5f97\u5230\u4efb\u610f\u7684 $$(1-x)^{-k}$$ \u5c55\u958b\u5f0f\u4fc2\u6578\u3002<\/p>\n

\u4f8b\u5982\uff1a<\/p>\n

$$(1-x)^{-2}$$ \u7684 $$x^2$$\u00a0\u9805\u4fc2\u6578 $$3$$\uff0c\u4fbf\u662f\u4e0a\u4e00\u5217\u524d $$3$$ \u9805\u4e4b\u548c\uff0c\u5373 $$a_{22}=a_{10}+a_{11}+a_{12}$$<\/p>\n

$$(1-x)^{-3}$$ \u7684 $$x^4$$\u00a0\u9805\u4fc2\u6578 $$15$$\uff0c\u4fbf\u662f\u4e0a\u4e00\u5217\u524d $$5$$ \u9805\u4e4b\u548c\uff0c
\n\u5373 $${a_{34}} = {a_{20}} + {a_{21}} + {a_{22}} + {a_{23}} + {a_{24}}$$<\/p>\n

$$\\cdots$$<\/p>\n

$$(1-x)^{-k}$$\u00a0\u7684 $$x^n$$ \u9805\u4fc2\u6578\uff0c\u4fbf\u662f\u4e0a\u4e00\u5217\u524d $$n+1$$ \u9805\u4e4b\u548c\uff0c
\n\u5373 $${a_{kn}} = {a_{k – 1,0}} + {a_{k – 1,1}} + {a_{k – 1,2}} +\\ldots+ {a_{k – 1,n}}$$<\/p>\n

\"55540_c1\"<\/a>

\u8868\u4e00 \u4e8c\u9805\u5c55\u958b\u5f0f\u6307\u6578\u8ca0\u51aa\u6b21\u6642\u7684\u4fc2\u6578\u8868<\/p><\/div>\n

\u70ba\u4ec0\u9ebc\u4e0a\u8ff0\u898f\u5f8b\u6210\u7acb\u5462\uff1f\u4f8b\u5982\u4e0a\u8ff0 $$(1-x)^{-2}$$\u00a0\u7684 $$x^2$$ \u9805\u4fc2\u6578\u70ba $$3$$ \u7684\u539f\u56e0\u662f\u56e0\u70ba\uff1a<\/p>\n

$$\\begin{array}{ll}{(1 – x)^{ – 2}} &= {(1 – x)^{ – 1}}{(1 – x)^{ – 1}} \\\\&= (1 + x + {x^2} + {x^3} + {x^4} +\\cdots )(1 + x + {x^2} + {x^3} + {x^4} +\\cdots)\\end{array}$$<\/p>\n

\u5176\u4e2d\uff0c\u5c55\u958b\u5f8c\u7684 $$x^2$$\u00a0\u9805\uff0c\u662f\u4f9d\u6b21\u7531\u5de6\u62ec\u865f\u88e1\u7684 $$1$$ \u4e58\u4e0a\u53f3\u62ec\u865f\u88e1\u7684 $$x^2$$\u00a0\u52a0\u4e0a\u5de6\u62ec\u865f\u88e1\u7684 $$x$$\u00a0\u4e58\u4e0a\u53f3\u62ec\u865f\u88e1\u7684 $$x$$\uff0c\u518d\u52a0\u4e0a\u5de6\u62ec\u865f\u88e1\u7684 $$x^2$$\u00a0\u4e58\u4e0a\u53f3\u62ec\u865f\u88e1\u7684 $$1$$ \u800c\u4f86\u7684\u3002\u56e0\u6b64\u5176\u4fc2\u6578\u70ba $$1+1+1$$\u3002<\/p>\n

\u53c8\u6216\u8005\uff1a<\/p>\n

$$\\begin{array}{ll}(1-x)^{-3}&= {(1 – x)^{ – 2}}{(1 – x)^{ – 1}}\\\\&= (1 + 2x + 3{x^2} + 4{x^3} + 5{x^4} +\\cdots )(1 + x + {x^2} + {x^3} + {x^4} +\\cdots )\\end{array}$$<\/p>\n

\u5176\u4e2d\uff0c$$x^4$$\u00a0\u9805\u4fc2\u6578 $$15$$\uff0c\u662f\u4f9d\u6b21\u7531\u5de6\u62ec\u865f\u88e1\u7684 $$1$$ \u4e58\u4e0a\u53f3\u62ec\u865f\u88e1\u7684 $$x^4$$\uff0c\u52a0\u4e0a\u5de6\u62ec\u865f\u88e1\u7684 $$2x$$\u00a0\u4e58\u4e0a\u53f3\u62ec\u865f\u88e1\u7684 \u00a0$$x^3$$\uff0c\u52a0\u4e0a\u5de6\u62ec\u865f\u88e1\u7684 $$3x^2$$\u00a0\u4e58\u4e0a\u53f3\u62ec\u865f\u88e1 $$x^2$$\uff0c\u52a0\u4e0a\u5de6\u62ec\u865f\u88e1\u7684 $$4x^3$$\u00a0\u4e58\u4e0a\u53f3\u62ec\u865f\u88e1\u7684 $$x$$\uff0c\u6700\u5f8c\u52a0\u4e0a\u7531\u5de6\u62ec\u865f\u88e1 $$5x^4$$\u00a0\u4e58\u4e0a\u53f3\u62ec\u865f\u88e1\u7684 $$1$$ \u800c\u4f86\u7684\u3002<\/p>\n

\u56e0\u6b64\u5176\u4fc2\u6578\u70ba $$(1-x)^{-2}$$\u00a0\u4e4b\u524d $$5$$ \u9805\u4fc2\u6578\u548c $$1+2+3+4+5=15$$\u3002<\/p>\n

\u7531\u4e0a\u8ff0\u300c\u7121\u7aae\u591a\u9805\u5f0f\u300d\u4e4b\u4e58\u6cd5\uff0c\u4e0d\u96e3\u770b\u51fa\u6b64\u898f\u5f8b\u70ba\u4f55\u6703\u6210\u7acb\uff0c\u5229\u7528\u6b64\u898f\u5f8b\u4fbf\u53ef\u9020\u51fa\u6574\u5f35\u8868\u3002\u800c\u6b64\u8868\u4fbf\u76f8\u7576\u65bc\u4e8c\u9805\u5f0f\u5b9a\u7406\uff0c\u7576\u6307\u6578\u70ba\u8ca0\u6574\u6578\u6642\u7684\u5c55\u958b\u5f0f\u4fc2\u6578\u8868\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u4e5f\u7531\u65bc\u6c5f\u6236\u6642\u671f\u65e5\u672c\u6578\u5b78\u5bb6\u4e26\u672a\u767c\u5c55\u51fa\u73fe\u4ee3\u7b26\u865f\u7cfb\u7d71\uff0c\u56e0\u6b64\u7121\u6cd5\u8f03\u4e00\u822c\u6027\u5730\u5beb\u4e0b\u4e8c\u9805\u5c55\u958b\u5f0f\u3002\u5fc5\u9700\u85c9\u7531\u300c\u8868\u300d\u7684\u65b9\u5f0f\uff0c\u8a18\u9304\u5404\u9805\u4fc2\u6578\uff0c\u540c\u6642\u501f\u7528\u6b64\u8868\u4f86\u63a8\u5ee3\u76f8\u95dc\u898f\u5f8b\u3002<\/p>\n

\u6700\u5f8c\uff0c\u773c\u5c16\u7684\u8b80\u8005\u53ef\u80fd\u6703\u767c\u73fe\uff0c\u9019\u5f35\u8868\u4e0a\u7684\u6578\u5b57\u8207\u898f\u5f8b\uff0c\u4e8b\u5be6\u4e0a\u4fbf\u662f\u5927\u5bb6\u719f\u77e5\u7684\u300c\u5df4\u65af\u5361\u4e09\u89d2\u5f62\u300d\uff0c\u53ea\u4e0d\u904e\u5b83\u7684\u6392\u5217\u65b9\u5f0f\u662f\u4ee5\u5de6\u4e0a\u89d2\u4f5c\u70ba\u4e09\u89d2\u5f62\u7684\u9802\u9ede\u3002\u662f\u4e0d\u662f\u5f88\u795e\u5947\u5462\uff1f<\/p>\n

\u9023\u7d50\uff1a\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u63a8\u5ee3\uff08\u56db\uff09\uff1a\u548c\u7b97\u5bb6\u7684\u6578\u5b78\u8868\uff08\u4e0b\uff09<\/a><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u4e8c\u9805\u5f0f\u5b9a\u7406\u7684\u63a8\u5ee3\uff08\u4e09\uff09\uff1a\u548c\u7b97\u5bb6\u7684\u6578\u5b78\u8868\uff08\u4e0a\uff09 (The generalization of Binomial …<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[220,228,111],"tags":[3916,4048,6905,6903],"class_list":["post-55540","post","type-post","status-publish","format-standard","hentry","category-math03-01","category-math05","category-mathematics00","tag-3916","tag-4048","tag-6905","tag-6903","loop-entry","cat-220","cat-228","cat-111","no-thumbnail"],"views":4837,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/55540","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=55540"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/55540\/revisions"}],"predecessor-version":[{"id":86956,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/55540\/revisions\/86956"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=55540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=55540"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=55540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}