
嘻道奇闻
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高考必考!等差等比求和心方法总结:公式推导+易错点剖析
鍝庡懄鍠傦紒楂樿€冩暟瀛﹂噷锛屾暟鍒楁眰鍜屾槸涓嶆槸鎬昏浣犲ご鐤硷紵鐗瑰埆鏄湅鍒?绛夊樊脳绛夋瘮"杩欑缁勫悎棰樺瀷锛屾槸涓嶆槸鎰熻鑴戣鍡″棥鍝嶏紵鍒厡锛佷粖澶╁挶浠氨鎶婅繖鍧楃‖楠ㄥご鍟冧笅鏉ワ紝鎴戞墦鍖呯エ鐪嬪畬杩欑瘒鏂囩珷锛屼綘鑷冲皯鑳藉鎷?鍒嗭紒
涓€銆佸叕寮忔硶锛氬熀纭€涓殑鍩虹涓嶈兘涓?/h3>
鍜变滑鍏堟潵鐩樼洏鈥?strong>鈥嬬瓑宸暟鍒楀拰绛夋瘮鏁板垪鐨勫熀纭€鍏紡鈥?/strong>鈥嬶紝杩欏彲鏄墍鏈夊彉褰㈤鐨勫湴鍩哄晩锛佺瓑宸暟鍒楁眰鍜屽叕寮忛暱杩欐牱锛氣€?strong>鈥婼n = n(a鈧?+ a鈧?/2鈥?/strong>鈥嬶紝璇寸櫧浜嗗氨鏄椤瑰姞鏈」涔橀」鏁板啀闄や互2銆傜瓑姣旀暟鍒楃殑姹傚拰鍏紡绋嶅井澶嶆潅鐐癸細褰撳叕姣攓鈮?鏃讹紝鈥?strong>鈥婼n = a鈧?1 - q鈦?/(1 - q)鈥?/strong>鈥嬨€傝繖涓や釜鍏紡灏卞儚鏁板鐣岀殑"绛峰瓙鍏勫紵"锛屽垎寮€鐢ㄧ畝鍗曪紝缁勫悎璧锋潵濞佸姏鏃犵┓銆?/p>
涓句釜鏍楀瓙馃尠锛氱瓑宸暟鍒?銆?銆?...鍓?0椤瑰拰锛岀洿鎺ュ鍏紡锛?2+29)脳10/2=155銆傜瓑姣旀暟鍒?銆?銆?2...鍓?椤瑰拰锛岀敤鍏紡3脳(1-2鈦?/(1-2)=93銆傛槸涓嶆槸姣旀兂璞′腑绠€鍗曪紵
鈥?strong>鈥嬫槗閿欑偣棰勮鈥?/strong>鈥嬶細杩欓噷鏈変釜鍧戣娉ㄦ剰锛佸緢澶氫汉绠楃瓑姣旀暟鍒楁椂锛岀湅鍒皅=1灏辩洿鎺ョ敤Sn=na鈧侊紝浣嗛亣鍒伴鐩"鍏瘮涓嶄负1鐨勭瓑姣旀暟鍒?鏃讹紝鍗冧竾璁板緱楠岃瘉q=1鐨勬儏鍐垫槸鍚﹀瓨鍦ㄣ€?/p>
浜屻€侀敊浣嶇浉鍑忥細灏忓績鍒鍚嶅瓧鍚撳埌
杩欎釜鏂规硶涓撴不鍚勭鈥?strong>鈥嬬瓑宸楃瓑姣斺€?/strong>鈥嬬殑鐤戦毦鏉傜棁锛屾瘮濡傞€氶」鏄痭路2鈦胯繖绉嶇被鍨嬨€傛搷浣滃彛璇€灏卞洓姝ワ細鈥?strong>鈥嬩竴鍐欎簩涔樹笁鍑忓洓鏁寸悊鈥?/strong>鈥嬨€傚叿浣撴€庝箞鐜╋紵鍜变滑鐢ㄤ釜渚嬮鎷嗚В锛?/p>
姹傛暟鍒梴n路3鈦縸鐨勫墠n椤瑰拰銆傚厛鍐欏嚭Sn=1脳3 + 2脳3虏 +...+n脳3鈦匡紝鐒跺悗涓よ竟鍚屼箻鍏瘮3寰楀埌3Sn銆傛帴鐫€鎶婁袱涓紡瀛愪笂涓嬪榻愮浉鍑忥紝绁炲鐨勪簨鎯呭彂鐢熶簡鈥斺€斾腑闂撮」鍏ㄩ儴鎶垫秷锛佹渶鍚庢暣鐞嗗緱Sn=(3/4)[(2n-1)3鈦?+1]銆?/p>
鈥?strong>鈥嬭娉暀璁€?/strong>鈥嬶細鍘诲勾楂樿€冨氨鏈?5%鐨勮€冪敓鍦ㄨ繖閲岀炕杞︼紒甯歌閿欒鍖呮嫭锛氬繕璁板榻愰」鏁般€佺鍙锋悶鍙嶃€佹渶鍚庢暣鐞嗘椂绯绘暟绠楅敊銆傚缓璁敤涓嶅悓棰滆壊绗旀爣娉ㄥ榻愰」锛屽仛瀹岄┈涓婄敤n=1銆?浠e叆楠岃瘉銆?/p>
涓夈€佽椤圭浉娑堬細鍖栭缁垫帉鎷嗚В闅鹃
杩欎釜鏂规硶鐨勭簿楂撳氨鍍忔媶涔愰珮锛屾妸閫氶」鎷嗘垚鍓嶅悗鐩稿噺鐨勫舰寮忋€備妇涓粡鍏告渚嬶細姹?/(1脳2)+1/(2脳3)+...+1/[n(n+1)]鐨勫拰銆傞€氳繃瑁傞」鍙樻垚(1 - 1/2)+(1/2 -1/3)+...锛屼腑闂寸殑椤瑰叏娑堜簡锛屾渶鍚庡彧鍓? - 1/(n+1)銆?/p>
鈥?strong>鈥嬮噸鐐瑰彛璇€鈥?/strong>鈥嬶細閬囧埌鍒嗗紡鍏堢湅鍒嗘瘝鑳戒笉鑳芥媶锛佸父瑙佽椤瑰叕寮忚鐗㈣繖鍑犱釜锛?/p>
- 1/[n(n+k)] = (1/k)(1/n - 1/(n+k))
- 1/(鈭歯 +鈭?n+1)) = 鈭?n+1) -鈭歯
- n路2鈦?= (An+B)2鈦?-[A(n-1)+B]2鈦库伝鹿 锛堢瓑宸楃瓑姣斾笓鐢級
鍥涖€佷竾鑳藉叕寮忥細鑰冨満鏁戝懡绋昏崏
杩欎釜鏂规硶绠€鐩村氨鏄綔寮婄鍣紒瀵逛簬绛夊樊脳绛夋瘮鏁板垪锛岀洿鎺ュ鐢ㄢ€?strong>鈥婼n=(An+B)q鈦?- B鈥?/strong>鈥嬨€傛瘮濡傛眰閫氶」鏄?2n+1)3鈦跨殑鏁板垪鍜岋紝璁維n=(An+B)3鈦?- B銆傚甫鍏=1寰桽鈧?3=(A+B)3 - B锛宯=2寰桽鈧?3+21=24=(2A+B)9 - B銆傝В鏂圭▼缁勫緱A=2銆丅=-3锛屾墍浠n=(2n-3)3鈦?+3銆?/p>
鈥?strong>鈥嬩娇鐢ㄩ』鐭モ€?/strong>鈥嬶細铏界劧杩欎釜鏂规硶蹇噯鐙狅紝浣嗗ぇ棰樻厧鐢紒寤鸿鍏堢敤杩欎釜鏂规硶蹇€熷緱鍑虹瓟妗堬紝鍐嶇敤閿欎綅鐩稿噺娉曞啓杩囩▼锛屽弻閲嶄繚闄╀笉涓㈠垎銆?/p>
浜斻€佷釜浜虹鎴跨绫?/h3>
鏍规嵁鎴戝崄骞村甫姣曚笟鐝殑缁忛獙锛岀粰鍚勪綅涓変釜閿﹀泭锛?/p>
- 鈥?strong>鈥嬮敊棰樹笁闂硶鈥?/strong>鈥嬶細涓轰粈涔堥敊锛熷摢涓楠ら敊锛熶笅娆℃€庝箞闃诧紵
- 鈥?strong>鈥嬬壒娈婂€奸獙璇佲€?/strong>鈥嬶細绠楀畬鍜岀珛鍒讳唬鍏=1銆?楠岃瘉
- 鈥?strong>鈥嬫椂闂村垎閰嶁€?/strong>鈥嬶細5鍒嗛挓娌℃€濊矾椹笂璺筹紝鍋氬畬澶ч鍐嶅洖鏉?/li>
鏈€鍚庤鍙ユ帍蹇冪獫鐨勮瘽锛氭暟鍒楁眰鍜岀湡涓嶉毦锛屽叧閿槸鎺屾彙濂楄矾澶氱粌涔犮€傛垜瑙佽繃澶瀛︾敓浠?0鍒嗙粌鍒版弧鍒嗙殑閫嗚锛屼粬浠渶澶х殑璇€绐嶅氨鏄€斺€旀瘡澶╁潥鎸佸仛3閬撴暟鍒楅锛岄浄鎵撲笉鍔紒浣犲彲鑳戒細鎯?姣忓ぉ3閬撳お灏戜簡鍚э紵"鍢匡紝閲嶇偣鍦ㄥ潥鎸侊紒灏卞儚瀛橀挶锛屾瘡澶╁瓨3鍧楋紝楂樿€冩椂浣犲氨鏄暟瀛︾晫鐨?鍦熻豹"锛?/p>