1、1-2-2-1.分?jǐn)?shù)裂項(xiàng).題庫(kù)教師版page1of18目W W蚱教學(xué)目標(biāo)本講知識(shí)點(diǎn)屬于計(jì)算大板塊內(nèi)容，其實(shí)分?jǐn)?shù)裂項(xiàng)很大程度上是發(fā)現(xiàn)規(guī)律、利用公式的過(guò)程，可以分為觀察、改造、運(yùn)用公式等過(guò)程。很多時(shí)候裂項(xiàng)的方式不易找到，需要進(jìn)行適當(dāng)?shù)淖冃危蛘呦冗M(jìn)行一部分運(yùn)算，使其變得更加簡(jiǎn)單明了。本講是整個(gè)奧數(shù)知識(shí)體系中的一個(gè)精華部分，列項(xiàng)與通項(xiàng)歸納是密不可分的，所以先找通項(xiàng)是裂項(xiàng)的前提，是能力的體現(xiàn)，對(duì)學(xué)生要求較高。型運(yùn)算將算式中的項(xiàng)進(jìn)行拆分，使拆分后的項(xiàng)可前后抵消，這種拆項(xiàng)計(jì)算稱(chēng)為裂項(xiàng)法.裂項(xiàng)分為分?jǐn)?shù)裂項(xiàng)和整數(shù)裂項(xiàng)，常見(jiàn)的裂項(xiàng)方法是將數(shù)字分拆成兩個(gè)或多個(gè)數(shù)字單位的和或差。遇到裂項(xiàng)的計(jì)算題時(shí)，要仔細(xì)的觀察每項(xiàng)

2、的分子和分母，找出每項(xiàng)分子分母之間具有的相同的關(guān)系，找出共有部分，裂項(xiàng)的題目無(wú)需復(fù)雜的計(jì)算，一般都是中間部分消去的過(guò)程，這樣的話，找到相鄰兩項(xiàng)的相似部分，讓它們消去才是最根本的。(1)對(duì)于分母可以寫(xiě)作兩個(gè)因數(shù)乘積的分?jǐn)?shù)，即，形式的，這里我們把較小的數(shù)寫(xiě)在前面，即ab,ab那么有，(I1)abbaab(2)對(duì)于分母上為3個(gè)或4個(gè)連續(xù)自然數(shù)乘積形式的分?jǐn)?shù)，即:形式的，我們有:n(n1)(n2)(n3)11r11in(n1)(n2)2n(n1)(n1)(n2)111n(n1)(n2)(n3)3n(n1)(n2)裂差型裂項(xiàng)的三大關(guān)鍵特征：(1)分子全部相同，最簡(jiǎn)單形式為都是1的，復(fù)雜形式可為都是x(x

3、為任意自然數(shù))的，但是只要將x提取出來(lái)即可轉(zhuǎn)化為分子都是1的運(yùn)算。(2)分母上均為幾個(gè)自然數(shù)的乘積形式，并且滿足相鄰2個(gè)分母上的因數(shù)“首尾相接”(3)分母上幾個(gè)因數(shù)間的差是一個(gè)定值。、“裂和”型運(yùn)算:常見(jiàn)的裂和型運(yùn)算主要有以下兩種形式:分?jǐn)?shù)裂項(xiàng)計(jì)算且班1 1住知識(shí)點(diǎn)撥1n(n1)(n2)1(n1)(n2)(n3)【解析】本題為典型的“隱藏在等差數(shù)列求和公式背后的分?jǐn)?shù)裂差型裂項(xiàng)”的項(xiàng)開(kāi)始入手，通過(guò)公式的運(yùn)算尋找規(guī)律。從第一項(xiàng)開(kāi)始，對(duì)分母進(jìn)行等差數(shù)列求和運(yùn)算公式的1(11)1原式例31011(1)abab1abababb22(2)a_b_ab2baabb裂和型運(yùn)算與裂差型運(yùn)算的對(duì)比：裂差型運(yùn)算的核

4、心環(huán)節(jié)是“兩兩抵消達(dá)到簡(jiǎn)化的目的”同時(shí)還有轉(zhuǎn)化為“分?jǐn)?shù)湊整”型的，以達(dá)到簡(jiǎn)化目的。,裂和型運(yùn)算的題目不僅有“兩兩抵消”型的,例1122【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【關(guān)鍵詞】美國(guó)長(zhǎng)島,一，1【解析】原式11例題精講113-4T-55【難度】小學(xué)數(shù)學(xué)競(jìng)賽提醒學(xué)生注意要乘以（分母差鞏固1011【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)11112原式12109【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)III原式110例2715【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)162星）分之一,如改為:【題型】計(jì)算111335，計(jì)算過(guò)程就要變?yōu)?577915960【難度】1石）2:43【難度】9日2星1(59【題型】計(jì)算111106012【題型】計(jì)算21-13107_15100【難度】3M【題型】計(jì)

5、算問(wèn)題。此類(lèi)問(wèn)題需要從最簡(jiǎn)單(12)2mmIII991011-2-2-1.分?jǐn)?shù)裂項(xiàng).題庫(kù)100101教師版(11、200.99)1-101101101page2of18【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析工工1335【答案】5010115T1III計(jì)算:25【難度】2星991012(1【題型】計(jì)算1111_359910150101【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【關(guān)鍵詞】迎春杯,二川【難度】2星2325【題型】計(jì)算原式25初賽,1I2六年級(jí)11325125-2252524c12251225148【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【關(guān)鍵詞】臺(tái)灣，2518122511216III251251200020042004原式211532計(jì)算:【難度】

6、2星小學(xué)數(shù)學(xué)競(jìng)賽，初賽2008【題型】計(jì)算251162511625116【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)原式例41250150250132六III,2115-327115111650015015011502III15011502162272229129【難度】3星1111711111616【題型】計(jì)算1111222229292計(jì)算:11111824488012011681224【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【關(guān)鍵詞】101中學(xué)【難度】2星1)128288【題型】計(jì)算解析原式III-)12824466816181/111122446III1161、一)128189644289【鞏固】12工工工工工工6122030425672

7、90【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】2星【題型】計(jì)算1-2-2-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)教師版page3of18【關(guān)鍵詞】走美杯，初賽，六年級(jí)【解析】根據(jù)裂項(xiàng)性質(zhì)進(jìn)行拆分為：11111111612203042567290111111112-34T5566y7-889910_11_221051(2110110【鞏固】工工工，104088154238【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】3星5【鞏固】1111L工工3610152128【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】6星【關(guān)鍵詞】走美杯，6年級(jí)，決賽【題型】計(jì)算【解析】原式1112III11112-2233III【鞏固】計(jì)算：112612203042567290【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度

8、】3星【關(guān)鍵詞】走美杯，6年級(jí)，決賽【題型】計(jì)算111111111、解析】原式()223344556677889910111112(2334III【解析】原式111811111414171-2-2-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)教師版page4of18【題型】計(jì)算【答案】534【例5】計(jì)算:III1200120032005【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【題型】計(jì)算【關(guān)鍵詞】華杯賽，總決賽，二試一，1【解析】原式141135III2001200320032005100400320032005120480451004003120480454.506例6區(qū)113-3.753.23【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【關(guān)鍵詞】仁華學(xué)校1工工二31

9、53563【難度】3星【解析】原式1821619011331.253714611213-12230.8【題型】計(jì)算111355779117946318_232442936【例7】計(jì)算：112126【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【關(guān)鍵詞】小數(shù)報(bào)，初賽1134-1220【難度】【題型】計(jì)算【解析】原式123Ml201111L2612204202102102102021021111122331212021021III120212021【鞏固】計(jì)算：200812009201011201112012=1854108180270【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】2星【題型】 計(jì)算【答案】1-2-2-1.分?jǐn)?shù)裂項(xiàng).題庫(kù)教師版pag

10、e5of18【關(guān)鍵詞】學(xué)而思杯，6年級(jí)，1試11【解析】原式20082009201020112012366911111.112010591223561005054【答案】10050&54【鞏固】計(jì)算：11A。26153577【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】2星【題型】計(jì)算【關(guān)鍵詞】學(xué)而思杯，6年級(jí)解析原式1325272261535771111111112233557711110III 11【鞏固】計(jì)算：1工工工工工工315356399143195【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】3星【解析】分析這個(gè)算式各項(xiàng)的分母，可以發(fā)現(xiàn)它們可以表示為：322113,1542135,19514211315,所以原式一1

11、1112131112115【答案】-15【鞏固】計(jì)算：1526【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【關(guān)鍵詞】四中1【解析】原式11299-19919911_1_1_335577991111123521371511192997011220309702【難度】3星161112III223991001111工2239910011001-2-2-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)11111111313151159899.9900【題型】計(jì)算9900教師版page6of1811912121511518【題型】計(jì)算11n1n11112n1nn12n1nnn1原式1212231112128935144【答案】里144494919800【解析原

12、式=1+1+1+1+113535719212324620222428160+1046534003234003238625340032【答案】港534003244939597959799【難度】3星【題型】計(jì)算解析()()1335355711320011111()()93959597959797991397999603100例8198100123【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【題型】計(jì)算【鞏固】計(jì)算：一1一123【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析】原式1（二212/HI【難度】112323198991003星113434【題型】計(jì)算19899991001494949492990019800【解析】首先分析出11788-9

13、【鞏固】計(jì)算:135【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)III1202224【題型】計(jì)算.)+1(，2123424-)222440+654832112鞏固135【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)1-2-2-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)教師版page7of18例9IIIII1234【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析】原式1-311111678978910【題型】計(jì)算23453456【難度】3星III11223423432009603鞏固99123【考點(diǎn)】 分?jǐn)?shù)裂項(xiàng)解析99123982349734519897123434599100101【難度】3星【題型】計(jì)算1001=100_1=100112312323123231002=1002100_1234234

14、2 342343 41003=1003100_13453453 4534545100991910099100101245110199100101991001019910010199100101100101100100100123234345100991001011)10010111100(22110100)11()2101245110134578989101111193123891021601192160【例10】計(jì)算:【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析】如果式子中每一項(xiàng)的分子都相同，那么就是一道很常見(jiàn)的分?jǐn)?shù)裂項(xiàng)的題目.但是本題中分子不相同，而是成等差數(shù)列，且等差數(shù)列的公差為2.相比較于2,4,6,這一公

15、差為2的等差數(shù)歹U（該數(shù)列的第n個(gè)數(shù)恰好為n的2倍），原式中分子所成的等差數(shù)列每一項(xiàng)都比其大3,所以可以先把原式中每一項(xiàng)的分子都分成3與另一個(gè)的和再進(jìn)行計(jì)算.原式III3168910III1-2-2-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)8910III88910教師版page8of18鞏固3-12342171819203星【題型】計(jì)算【解析】原式111113c(3123234234345113192011139171819181920)123【題型】計(jì)算【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)6840181920181920【答案】但68401127-22III89910211.I，123491012910III119101122901

16、102346015也可以直接進(jìn)行通項(xiàng)歸納.根據(jù)等差數(shù)列的性質(zhì)，可知分子的通項(xiàng)公式為2n3nn1n2n13一.一一,再將每一項(xiàng)的n2nn1n2-分別加在一起進(jìn)行裂項(xiàng).后面的過(guò)程與前面的方法相同.1n2【答案】竺15【鞏固】計(jì)算:/51155(III1710【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】3星【關(guān)鍵詞】迎春杯，初賽，五年級(jí)-）91011【題型】計(jì)算【解析】本題的重點(diǎn)在于計(jì)算括號(hào)內(nèi)的算式:III178910一-一.這個(gè)算式不同91011于我們常見(jiàn)的分?jǐn)?shù)裂項(xiàng)的地方在于每一項(xiàng)的分子依次成等差數(shù)列，而非常見(jiàn)的分子相同、或分子是分母的差或和的情況.觀察可知523,所以應(yīng)當(dāng)對(duì)分子進(jìn)行適當(dāng)?shù)淖冃危怪D(zhuǎn)化成我們熟悉的形

17、式.34,即每一項(xiàng)的分子都等于分母中前兩個(gè)乘數(shù)的和，所以523423231341HI171989109109101143124145III91HI10III111101111111011111131311所以原式川11011111115521031311833158331113155一651.55（法二）上面的方法是最直觀的轉(zhuǎn)化方法，但不是唯一的轉(zhuǎn)化方法.由于分子成等差數(shù)列,通項(xiàng)公式為and,其中d為公差.如果能把分子變成這樣的形式，再將變成兩個(gè)分?jǐn)?shù)，接下來(lái)就可以裂項(xiàng)了.而等差數(shù)列的a與nd分開(kāi)，每一項(xiàng)都523412223413V7HI132I345I22III1789101821991011

18、192891091011IIIIII1-2-2-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)891028910910118910910113445教師版III9109101121011page9of181112-3-4III910101112231011111122341220311422031一,55所以原式115531651.55（法三）本題不對(duì)分子進(jìn)行轉(zhuǎn)化也是可以進(jìn)行計(jì)算的:451341719891091011III17萬(wàn)萬(wàn)18919萬(wàn)萬(wàn)910101113-4III1917萬(wàn)萬(wàn)萬(wàn)萬(wàn)191910210113445119311991021011123所以原式1022055311155651.55（法四）對(duì)于這類(lèi)變化較

19、多的式子，最基本的方法就是通項(xiàng)歸納.先找每一項(xiàng)的通項(xiàng)公式:2n1ann(n1)(n如果將分子2n【答案】651(n2,3,，9)2）1分成2n和1,就是上面的法二；如果將分子分成n和n1,就是上面的法【鞏固】計(jì)算：3124【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)5人m【難度】3星【題型】計(jì)算【解析】觀察可知原式每一項(xiàng)的分母中如果補(bǔ)上分子中的數(shù)，就會(huì)是將每一項(xiàng)的分子、分母都乘以分子中的數(shù).即：5個(gè)連續(xù)自然數(shù)的乘積，所以可以先原式324252123452345634567Ml1221011121314現(xiàn)在進(jìn)行裂項(xiàng)的話無(wú)法全部相消，需要對(duì)分子進(jìn)行分拆，考慮到每一項(xiàng)中分子、分母的對(duì)稱(chēng)性,可以用平方差公式:原式3232154,

20、4242-2264,55212345154123451234562642345611234345456122it1112131234523456345671011121314101441011121314II10111；131411121213123423452345345610111213111213141-2-2-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)教師版page10of18616【例iii1111122312131 2341112131411111771111175122121324111213148111213148211148308616223【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析】原式122212112【難度】4星

21、1-2答案3628799口水362880031212141334M91023436287993628800III101III【例12,2121233123410題型】計(jì)算92341H1010134UI91234口|910【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析】原式1215039504015040【難度】4星415123456【題型】計(jì)算61503950401III2345612345671【鞏固】計(jì)算：23M幽.3!4!100!【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】4星【題型】計(jì)算【解析】原式為階乘的形式，較難進(jìn)行分析，但是如果將其寫(xiě)成連乘積的形式，題目就豁然開(kāi)朗了.原式23123123431411231234111III

22、III121231231112123100199123UI1001001123UI100111234123M991JL2100!1123|100【例132100!1(12)3(12)(123)4HI50(123)(1234)(123|49)(12|50)1-2-2-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)教師版page11of18【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】3星【題型】計(jì)算【解析】 原式=127412751(12)【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)21(12)3)(12)(11121004610+101523)【難度】(123)3星3(12)(123)(12I99)(12I1100)原式1-111505012川1005049505050

23、495050211(12)(132)(123)III【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】2星【解析】原II1551555012251275122512741275(1234)100299)(12*100)1299【題型】計(jì)算12311201，所以100(1234555145；15510|9)(123m10)【題型】計(jì)算【例1431【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【關(guān)鍵詞】仁華學(xué)校152117211921【難度】111213星11321【題型】計(jì)算這題是利用平方差公式進(jìn)行裂項(xiàng):b21(2(2314)()(4466111114-)14668314計(jì)算:(1122)1(12)3(1【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)1-

24、2-2-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)-)8110(8101(a1b)101101212-)1211(ab),1(-1212一)14(1III(114?)(1【難度】3星【題型】計(jì)算教師版page12of18111111132435981001999900【解析】1(12)i(12)(113)(13)所以,原式III48504949504925492549【鞏固】計(jì)算：12【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)22【解析】原式22!122222222233478【難度】3星222222324387222TZT2722-2233478【題型】計(jì)算2【鞏固】計(jì)算：331【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)22225171199311995152172

25、1199321199521【難度】3星【題型】計(jì)算一一，22【解析】原式1二1二-315121995【答案】9979979979979971996計(jì)算:12224246III2199419961112m2446I112199699711994119969971996【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)2Q2【解析21103-2.2_22435321421【難度】3星-2,2222420352)23184198210029921【題型】計(jì)算3415由于？42320434一,一81515可見(jiàn)原式422142-31III29912984243598100198475149501-2-2-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)教師版page

26、13of181961962111299100196292Q22鞏固計(jì)算考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析】式子中每一項(xiàng)的分子與分母初看起來(lái)關(guān)系不大,【題型】計(jì)算但是如果將其中的分母根據(jù)平方差公式分別變?yōu)椤敬鸢浮?2更1013573445475667222,1001,可以發(fā)現(xiàn)如果分母都加上1,那么恰好都是分子的4倍,所以可以先將原式乘以4后進(jìn)行計(jì)算，得出結(jié)果后除以4就得到原式的值了.原式1224262川1002422142162110021111212122142111002111150133557口9910199101150211-101150504101C63121011-2-2

27、-1,分?jǐn)?shù)裂項(xiàng).題庫(kù)【難度】3星【例15】5656【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)99107889910【難度】3星解析u口su56677889910【答案】-10【題型】計(jì)算1111(-)-(-)91011510旦10【鞏固】3651包U史57612203042【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【難度】3星【關(guān)鍵詞】第三屆，祖沖之杯，人大附中解析原式=36Q-U35723344556【答案】4【題型】計(jì)算673611116757233411.=467【鞏固】計(jì)算:【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)1325791011193457820212435【難度】3星【題型】計(jì)算【解析】原式1925731111322111115345784537791

28、117251220283042【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析】原式【難度】3星123111111【題型】計(jì)算13教師版page14of1811112123131113366555777444【鞏固】3549637791W561220304256【鞏固】11L工工巴上空巴233031415111912012327【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析】原式【難度】3星11111111124【題型】計(jì)算3303141311111117217717430341431【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析】原式【難度】3星57911131561220304256【題型】計(jì)算371-88111111rlic78233478811282111【答案】1011788810【鞏固】計(jì)算：156【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)79111315171912203042567290【難度】3星【解析】原式1U3U233445566711111111()()()(-233445561132105【題型】計(jì)算7889910788991011111111)()()()()677889910【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)【解析】原式21【例1617981751220153012【難度】3星【題型】計(jì)算1111111121114531外1一5一56222【考點(diǎn)】分?jǐn)?shù)裂項(xiàng)22221