四年級奧數(shù)等差數(shù)列二教師版

1、第五講等差數(shù)列二解題方法某些問題以轉(zhuǎn)化為求假設(shè)干個(gè)數(shù)的和解決這些問題時(shí)先要判斷這些數(shù)是否成為等差數(shù) 列,如果是等差數(shù)列才可以運(yùn)用它的一些公式.在解決自然數(shù)的數(shù)字問題時(shí),應(yīng)根據(jù)題目的具體特點(diǎn),有時(shí)可考慮將題中的數(shù)適當(dāng)分 組,并將每組中的數(shù)合理配對,使問題得以順利解決.例題1小王看一本書第一天看了 20頁,以后每天都比前一天多看 2頁,第30天看了 78頁正好看完.這本書共有多少頁提示根據(jù)條件“以后每天比前一天多看2頁可以知道他每天看的頁數(shù)都是根據(jù)一定規(guī)律排列的數(shù), 即20、22、24、76、78.要求這本書共有多少頁也就是求出這列數(shù)的和.解：由題意可知,這列數(shù)是一個(gè)等差數(shù)列,首項(xiàng) =20,末項(xiàng)=

2、78,項(xiàng)數(shù)=30,所以這本書共有 20+78 X 30+2=1470 頁答：這本書共有1470頁.引申1、文麗學(xué)英語單詞,第一天學(xué)會了 3個(gè),以后每天都比前一天多學(xué)會 1個(gè),最后一天學(xué)會了 21 個(gè).文麗在這些天中共學(xué)會了多少個(gè)英語單詞解：文麗每天學(xué)會的單詞個(gè)數(shù)是一個(gè)等差數(shù)列,即3、4、5、6、21.首項(xiàng)=3,末項(xiàng)=21,項(xiàng)數(shù)=21-3 +2+1=10.所以,文麗在這些天中共學(xué)會了3+21 X 10+ 2=120個(gè)答：文麗在這些天中共學(xué)會了120個(gè)英語單詞.2、李師傅做一批零件,第一天做了 25個(gè),以后每天都比前一天多做 2個(gè),第20天做了 63個(gè) 正好做完.這批零件共有多少個(gè)答：25+63

3、X 20+ 2=880 個(gè)3、小李讀一本短篇小說,她第一天讀了20頁這個(gè)等差數(shù)列共有多少項(xiàng) 答：這個(gè)等差數(shù)列共有29項(xiàng).例題2建筑工地上堆著一些鋼管如以下圖,求這堆鋼管一共有多少根.oooocboooo提示：根據(jù)圖可以知道,這是一個(gè)以3為首項(xiàng),以1為公差的等差數(shù)列,求鋼管一共有多少根其實(shí)是求 這列數(shù)的和.解：求鋼管一共有多少根 ,其實(shí)就是求3+4+5+9+10的和.項(xiàng)數(shù)=10-3 +1+1=8,根據(jù)公式求和為：3+4+5+9+10=3+10 X8 + 2=13X8+ 2=52根.答：這堆鋼管一共有52根.引中1、一些同樣粗細(xì)的圓木,像如以下圖一樣均勻地堆放在一起,最下面一層有70根.一共有多少

4、根圓木答案：2485根.2、用3根等長的火柴棍擺成一個(gè)等邊三角形,用這樣的等邊三角形,按以下圖所示鋪滿一個(gè)大的等邊三角形,如果這個(gè)大的等邊三角 形的底邊能放10根火柴棒,那么這個(gè)大的等邊三角形中一共要放 多少根火柴棒解：如果把圖中最上端的一個(gè)三角形看做第一層,與第一層緊相連的3個(gè)三角形2個(gè)向上的三角形,一個(gè)向下的三角形看做第二層,那么這個(gè)圖中一共有10層三角形.不難看出,這10層三角形每層所需火柴棒根數(shù),自上而下依次為：3,6,9,3X10.它們成等差數(shù)列,且首項(xiàng)為3,公差為3,項(xiàng)數(shù)為10.求火柴的總根數(shù),也就是求這個(gè)等差數(shù)列各項(xiàng)的和.即：3+6+9+ +30=3+30 X 10 + 2=33

5、 X 5=165卞艮答：這個(gè)大的等邊三角形中一共要放16升艮火柴棒.3、用相同的小立方體擺成如以下圖的形狀,如果共擺成10層,那么最下面有多少個(gè)小立方體答案：55個(gè)例題3有50把鎖的鑰匙搞亂了 ,為了使每把鎖都配上自己的鑰匙,至多要試多少次提示：開第一把鎖時(shí),如果不湊巧,試了 4田巴鑰匙還不行,那所剩的一把就一定能把它翻開,即開第 一把鎖至多需要試4砍,同理,開第二把鎖至多需要48次,開第三把鎖至多需試47次,等翻開第4毋巴 鎖,剩下的最后一把不用試,一定能翻開.解：根據(jù)以上分析,可以把此題轉(zhuǎn)化為求一個(gè)等差數(shù)列的和,即 49+48+47+ +2+1=49+1 X49+ 2=1225次答：至多要

6、試1225次.引中1、有60把鎖的鑰匙搞亂了 ,為了使每把鎖都配上自己的鑰匙,至多試多少次解：59+58+57+2+1=(59+1) X 59+2=1770(次)2、有一些鎖的鑰匙搞亂了 ,至多要試28次,就能使每把鎖都配上自己的鑰匙.一共有幾 把鎖的鑰匙搞亂了 答：一共有8把鎖的鑰匙搞亂了 .3、一輛公共汽車有66個(gè)座位,空車出發(fā)后,第一站上一位乘客,第二站上兩位乘客,第三站上 三位乘客,依次類推,第幾站后,車上坐滿乘客答：第11站后,車上坐滿乘客.例題4四(1)班45位同學(xué)舉行一次同學(xué)聯(lián)歡會,同學(xué)們在一起一一握手,且每兩個(gè)人只能握一次手,同學(xué)們共握了多少次手提示：假設(shè)4游同學(xué)排成一隊(duì),第1

7、位同學(xué)一次與其他同學(xué)握手,一共握了 44次,第2位同學(xué)因與第1 位同學(xué)已握手,只需要與另外4啦同學(xué)握手,一共握了 43次,這卞¥第3位同學(xué)只需與另外的42位同學(xué)握手, 依次類推.握手的次數(shù)分別為：44,43,42,3,2,1 ,這樣應(yīng)用等差數(shù)列求和公式即可解答.解：根據(jù)以上分析,可以把此題轉(zhuǎn)化為求一個(gè)等差數(shù)列的和即 44+43+42+ -+3+2+1=(44+1) X 44+2=990 (次)答：同學(xué)們共握了 990次手.引中1、學(xué)校進(jìn)行書法大賽,每個(gè)選手都要和其他所有選手各賽一場.如果有16人參加比賽,一共要進(jìn)行多少場比賽答案：15+14+13+ - +3+2+1= (15+1)

8、X 15+2=120 (場)2、在一次元旦晚會上,一共有48位同學(xué)和5位老師,每一位同學(xué)或老師都要和其他同學(xué)握一 次手.那么一共握了多少次手解：根據(jù)題意,一共有 48+5=53 (人)參加了這次晚會.所以,一共握手的次數(shù)為： 52+51+50+ +3+2+1= (52+1) X 52+2=1378 (次)答：一共握了 1378次手.3、一次朋友聚會,大家見面時(shí)總共握手28次.如果參加聚會的人和其余的每個(gè)人只握手一次,問參加聚會的共有多少人解：設(shè)共有n人參加了聚會,由于要求參加聚會的人和其余的每個(gè)人只握手一次,所以 一共握手(n-1 ) + (n-2) + +2+1=nX ( n-1 ) +2,

9、由于共握手 28次,所以 nx ( n-1 ) + 2=28,即nx (n-1 ) =56.又由于n是正整數(shù),通過計(jì)算,可知 8X 7=56, n=8,所以參加聚會 的共有8人.Class:答：參加聚會的共有8人.Name:A.Morning !B.Hello !C.Hi !()2. Nice to see you again !A.How are you ?B.Nice to see you , too .C.How do you do ?()3.Good night,mom !A.Night !B.Good night !()4.How do you do ?A.How are you ?

10、B.Fine,thanks .()5.HowmanyC.Good evening .C.How do you do ?storybooksdoyouhave ?A.I have 10.B.I can see 10.()6.Do you have new teachers?A.Yes,we do .B.Yes,we don ' t.()7.Who'C.Thirty yuan.C.Yes,we have .syourartteacher ?A.Mr Zhu.B.Miss Zhu.C.He ' s tall.()8.What ' s he like?A.He'

11、 s tall and strong .B.Yes,he is.C.Mr Zhu.()9.Is your English teacher young?A.No,she isn ' t.B.Yes,she is .C.No,she is.()10.?Her name is Chen Jie.A.What' s your name ?B.What ' s she name ? C.What ' s her name ?()11.? I like Chinese,math and English.A.What classes do you like?B.What do

12、 you like?C.What are you like ?()12.? We have English and P.E.C.What do you have on Monday?()13.? It ' s Monday.A.What is it today ? B.What day is it today ?C.What day is today ?()14.? I watch TV and do my homework.A.What do you do ? B.What do you do in Mondays?C.What do you do on Sundays ?()15.

13、May I have a look ?A.Sure.Here you are .B.Look !C.Here you are .)24.Thisisapple.It()16.Our math teacher isCanada.A.fromB.inC.at()17.Ithree new teachers.A.hasB.amC.have()18.What ' sChinese teacher like ?A.youB.yourC.you ' re()19.My P.E.teacher isthin.A.tooB.soC.very()20.There aredays in a wee

14、k.A.sixB.sevenC.eight()21.There aremonth(月)in a year(年).A.tenB.elevenC.twelve()22.I often watch TVSaturday .A.onB.inC.at()23.I like P.E.I don ' t lime music.A.butB.andC.soisred apple.A.a , aB.an,anC.an, a()25.Whatdo you like ?A.classesB.classC.class ' s六年級英語測試題Class:Name:()1. How are you ?A.

15、Fine,thanks.B.Yes,it is.C.How are you()2. Nice to meet you !A.Fine,thank you.B.OK.C.Nice to meet you ,too()3. How do you go to school ?A.I go to Canada by plane. B.I go to school by bike.C.What about you()4.HowdoyougototheUSA ?A.I usually go to school by bus. B.I go to England by ship.C.I go by plan

16、e.()5.?My homeis near thepost office.A. Where is your home ? B.OK.C.See you then!()6. See you at 2 o' clock.A.See you then !B.The fifth floor.C.It ' s easy.()7.?You can go by the No.15bus.A.It ' s not far.B. How can I get to Zhongshan Park ?C.Sure.A.Next to the cinema.B.Thank you .C.You

17、' re welcome.()9. Excusehere ?A.Yes,there is .B.No,it()10.?office.A.Yes,there is .B.No,it()11.A. Where is the cinema ,please ?me ,isthere a cinema nears not far.C.It ' s near the post office.? It ' s near the posts not far.C. Where is the library? It ' s next to the hospital.B.Is it

18、far ?C.Go straight.()12. How can I get to the hospitalA.It ' s next to the hospital.B.You can go by the No.201 bus. C.Thank you.()13.Thank you .A.Thank you .B.OK.C.You' re welcome.()14. How can I get to the museum?A.Go straight.Then turn left.B.Thank you .C.It ' s east of the cinema .()1

19、5. Where is the post office ?A.Go straight.Then turn left.B.Thank you .C.It ' s east of the cinema .()16. What are you going to do thisevening ? A.I ' m going to the cinema . B.I go to school on foot.C.Yes,it is.()17. Is it far ?A.No,it is .B.Yes , it is .C.Yes,it isn ' t.()18.do you go

20、to school ?()19. I go to schoolbike.A.onB. byC.get()20. Can I gofoot ?A.byB.atC.on()21.How can Ito the post office ?A.nearB. getC.for()22.me.A. ExcuseB. HowC.next()23.birthday to you !A. AfterB.HappyC.First()24. The hospital isthe left.A. atB.inC.on()25.is the bookstore ?A.WhereB.HowC.When四年級英語測試題Cl

21、ass:Name:( )1.Good afternoon ! A.Hello! B.Hi ! C.Afternoon ! ( )2.Nice to see you again ! A.How are you ? B.Hello ! C.Nice to see you , too. ( )3.Good night,moom ! A.Good evening ! B.Good night ! C.Night ! ( )4.How are you ? A.Fine,thank you . B.I am 10. C.Nine . ( )5.How do you do ?A.Fine ,Thank yo

22、u . B.How are you ? C.How do you do ? ( )6.How old are you ?A.How are you ? B.I am 11. C.I have 11. ()7.? I have 23.A.How many books do you have ? B.How many books can you see? C.How many book do you have ? ()8.? I can see 6.A.How many lights do you have ? B.How many lights can you see ?C.How many l

23、ight can you see ?( )9.May I have a look ?A.Sure.Here you are . B.Look! C.Sure.Here are you .()10.? 50 yuan.A.How much is this schoolbag ? B.How many is this schoolbag ?C.How much are this schoolbag ?( )11.Where' s my seat ?A.It' s near the door. B.It' s on the door. C.It' s under th

24、e door.()12.Let ' s clean the desks and chairs.A.All right ! B.Hello ! C.It' s nice .()13.What' s this ?A.There is a board . B.It' s a bee. C.It' s bee.()14.? My name is Mike.A.Here are you ? B.What' s your name ? C.How do you do ?()15.Who ' s the inventor of paper ?A.Chi

25、nese people. B.Oh! Great ! C.Hello!()16.I a student. You a teacher.A.am ,am B.are, are C.am, are( )17.This a boy.His nameZhang Peng.A.is, is B.am , is C.is , are()18.Let clean the fish bowl.A.I B. meC.my()19.This isapple.It isred apple .A.an ,anB.a, aC.an, a()20.We a new classroom.A.areB.haveC.has( )21.There a bee in our classroom.A. is B.are C.am()22. have