五年级奥数专题13：数列的求和

十三 数列的求和(A) 年级 班 姓名 得分 一、填空 1. 1～1991这1991个自然数中,所有的奇数之和与所有的偶数之和的差是______. 2. 计算: 1-3+5-7+9-11+…-1999+2001=______. 3. 计算: 100+99+98-97-96+95+94+93-92-91+…+10+9+8-7-6+5+4+3-2-1=______. 4. 计算: 1992+ -1 +2 -3 +4 -5 +…+1990 -1991 =______. 5. 100与500之间能被9整除的所有自然数之和是______. 6. 如左下图,一个堆放铅笔的 形架的最下层放1支铅笔,往上每一层都比它下面一层多放一支,最上面一层放120支.这个 形架上共放了______支铅笔. 7. 一堆相同的立方体堆积如下图所示.第一层1个,第二层3个,第三层6个,……,第10层有______个立方体. 8. 下面数列中各数呈现一定规律,其中第五项是几? 1,2,5,10,( ),26,37…. 9. 数列: 5.01, 6.02, 7.01, 5.02, 6.01, 7.02, …前20项的和是______. 10. 计算: . 二、解答题 11. 如下图,三角形每边2等分时,顶点向下的小三角形有1个;每边4等分时,顶点向下的小三角形有6个;每边10等分时,顶点向下的小三角形有几个? 20等分呢? 12. 计算: 13. 求值: 14. 求1991个自然数,其中一个是1991,使它们的倒数之和恰好为1(这些自然数不都相同). 十三 数列的求和(B) 年级 班 姓名 得分 一、填空题 1. 计算: (3+4+5+6+7+8+9+10+11+12+13+14+15)÷13=______. 2. 计算: . 3. 计算:(1+ )+(1+ ×2)+(1+ ×3)+…+(1+ ×10)+(1+ ×11)=______. 4. 在1,4,7,10,13,…,100中,每个数的前面加上一个小数点以后的总和等于______. 5. , 这239个数中所有不是整数的分数的和是______. 6. 计算: =______. 7. 计算: . 8. 计算: . 9. 计算:1+3 . 10. 把1到100的一百个自然数全部写出来,所用到的所有数码字的和是____. 二、解答题 11. 求: EMBED Equation.3 EMBED Equation.3 …+ . 12. 求: . 13. 求: 14. 一个家具厂生产桌的数目每个月增加10件,一年共生产了1920件,问这一年的12月份生产了多少件? ———————————————答 案—————————————————————— 答 案: 1. (1+3+…+1991)-(2+4+…+1990) =1+(3-2)+(5-4)+…+(1991-1990) =1+1+…+1 =996 2. 1-3+5-7+9-11+…-1999+2001 =1+(5-3)+(9-7)+(13-11)+…+(2001-1999) =1+2+2+…+2 =1001 3. 100+99+98-97-96+95+94+93-92-91+…+10+9+8-7-6+5+4+3-2-1 =100+(99-97)+(98-96)+95+(94-92)+(93-91)+…+10+(9-7)+(8-6)+5+(4 -2)+(3-1) =(100+95+…+10+5)+2+2+…+2 = =105×10+80 =1130 4. 1992+ -1 +2 -3 +4 -5 +…+1990 -1991 =[(2-1)+(4-3)+ …+(1992-1991)]+[( - )+( - )+ …+( - )] =996+996×( - ) =996+996× =996+166 =1162 5. 100到500之间9的倍数有9×12,9×13,…,9×55,共55-12+1=44个,它们的和是 =13266. 6. 型架上铅笔总数是 1+2+3+…+120= =7260(支). 7. 第一层有1个;第二层有1+2=3个;第三层有1+2+3=6个;…;第十层有 1+2+3+…+10= =55(个). 8. 这个数列相邻两项的差组成奇数数列: 1,3,5,7,9,11,…,故第五项是10+7=17. 9. 20÷3=6…2.前20项之和为 (5+6+7)×6+5+6+(0.01+0.02)×10=119.3 . 10. + + + + = + ×( - + - + - + - ) = + ×( - ) = + × = . 11. 三角形每边二、三、四等分后,每排所产生的顶角向下的小三角形的个数是1,2,3.同样,三角形每边10等分时,顶角向下的小三角形有 1+2+3+…+9= =45(个). 三角形每边20等分后,产生的顶角向下的小三角形有 1+2+3+…+19= =190(个). 12. =( - )× ; =( - )× ; ……………………………… =( - )× . 相加得 + +…+ = ( - ) = . 13. 1 +4 +7 +10 +13 +16 =(1+4+7+10+13+16)+( + + + + + ) = +( - + - +…+ - )× =51+( - )× =51 . 14. 因为 + + +…+ =1- + - + - +…+ - =1- . 所以 + + +…+ + =1. 1×2,2×3,3×4,…,1990×1991和1991这1991个自然数满足. ———————————————答 案—————————————————————— 答 案: 1. 解法一 (3+4+5+6+…+14+15)÷13 = ×13÷13 =9×13÷13 =9 解法二 (3+4+5+6+…+14+15)÷13 =[(3+10)+(4+9)+(5+8)+(6+7)+(11+15)+13+(12+14)]÷13 =13×9÷13 =9 2. + + +…+ = × ×1990 = =995.5 3. (1+ )+(1+ ×2)+(1+ ×3)+…+(1+ ×10)+(1+ ×11) =(1+1+…+1)+ ×(1+2+3+…+10+11) =11+ × =11+ ×6×11 =25. 4. 这列数的各个数是1,4,7,10,13,17,…,97,100.在每个数的前面加上小数点后,各个数的值都发生了变化.在这列数第1~3个数是一位数,每个数都缩小了10倍,第4个数到第33个数(10~97)是两位数,每个数都缩小了100倍,最后一个数100缩小了1000倍.先分别求出1,4,7的和以及第4个数到第33个数的和,再求出34个小数的和. 0.1+0.4+0.7+0.10+0.13+…+0.97+0.1 =(1+4+7)× +(10+13+…+97)× +0.1 =1.2+(10+97)÷2×30× +0.1 =1.2+16.05+0.1 =17.35 5. 从所有数的和中减去所有整数的和即为所有不是整数的分数的和.所以所求分数的和是 = = =2390-190 =2200. 6. = = = = 7. 仿上题,用裂项法解之. = = = 8. = = = = 9. 1+3 +5 +9 +11 +13 +15 +17 =(1+3+5+7+9+11+13+15+17)+( EMBED Equation.3 + + ) = =81+( ) =81+ =81 10. 把1到100的一百个自然数排成以下数阵 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 …… 90 91 92 93 94 95 96 97 98 99 100 如果100除外,从数阵中可以看出,99个数中,个位上0有9个,1到9九个数分别出现了十次;十位上的数字,1到9也分别出现了十次,最后一个数100,三个数码的和是1.所以所用到的所有数码字的和是 (1+2+3+…+8+9)×10×2+1 = ×9×10×2+1 =901 11. 和=1986×( - + - +…+ - ) =1986×( - ) =1986× = . 12. 和= + + +…+ = ×(11+21+31+…+81) = × = =4 . 13. 和= = =2 =2 =2 = . 14. 设1月份生产了 件,那么12月份生产了 +110件,一年共生产书桌 , 化简得 2 +110=320; 解得 =105. 所以12月份生产书桌105+110=215件. � EMBED PBrush ��� � EMBED PBrush ��� � EMBED PBrush ��� � EMBED PBrush ��� � EMBED PBrush ��� � EMBED PBrush ��� � EMBED PBrush ��� � EMBED PBrush ��� _987489329.unknown _987508026.unknown _987510080.unknown _987510828.unknown _991313800.unknown _996904451.unknown _997945410.unknown _991315202.unknown _991315111.unknown _987511193.unknown _987511367.unknown _991309220.unknown _987511287.unknown _987511308.unknown _987511088.unknown _987511179.unknown _987510976.unknown _987510305.unknown _987510443.unknown _987510723.unknown _987510427.unknown _987510125.unknown _987510186.unknown _987510124.unknown _987508756.unknown _987509561.unknown _987509725.unknown _987510028.unknown _987509679.unknown _987509527.unknown _987509560.unknown _987509559.unknown _987509351.unknown _987509454.unknown _987508446.unknown _987508689.unknown _987508727.unknown _987508513.unknown _987508330.unknown _987508411.unknown _987508277.unknown _987506287.unknown _987507661.unknown _987507882.unknown _987507921.unknown _987507948.unknown _987507896.unknown _987507775.unknown _987507806.unknown _987507716.unknown _987506844.unknown _987507097.unknown _987507484.unknown _987506914.unknown _987506620.unknown _987506699.unknown _987506326.unknown _987494440.unknown _987505857.unknown _987506114.unknown _987506144.unknown _987505885.unknown _987505482.unknown _987505696.unknown _987505428.unknown _987494013.unknown _987494097.unknown _987494201.unknown _987494215.unknown _987494051.unknown _987493896.unknown _987494011.unknown _987494012.unknown _987493946.unknown _987489721.unknown _987428830.unknown _987488034.unknown _987488810.unknown _987489251.unknown _987489263.unknown _987489208.unknown _987488484.unknown _987488682.unknown _987488199.unknown _987429129.unknown _987429697.unknown _987430697.unknown _987432413.unknown _987487179.unknown _987488004.unknown _987432438.unknown _987432477.unknown _987487153.unknown _987432458.unknown _987430887.unknown _987432270.unknown _987432336.unknown _987430839.unknown _987430725.unknown _987430777.unknown _987430712.unknown _987430435.unknown _987430581.unknown _987430667.unknown _987430696.unknown _987430436.unknown _987429895.unknown _987430255.unknown _987430321.unknown _987430434.unknown _987430284.unknown _987430142.unknown _987429913.unknown _987429829.unknown _987429605.unknown _987429653.unknown _987429446.unknown _987429546.unknown _987429545.unknown _987429364.unknown _987428932.unknown _987428962.unknown _987429031.unknown _987429091.unknown _987428950.unknown _987428907.unknown _987428922.unknown _987428885.unknown _987424458.unknown _987428232.unknown _987428748.unknown _987428802.unknown _987428827.unknown _987428794.unknown _987428654.unknown _987428747.unknown _987428746.unknown _987428579.unknown _987427615.unknown _987428041.unknown _987428081.unknown _987427964.unknown _987426676.unknown _987426705.unknown _987426667.unknown _987425511.unknown _987420006.unknown _987420268.unknown _987422558.unknown _987420254.unknown _987419975.unknown _987419991.unknown _987419932.unknown