范文一:分数数列求和
分数数列求和 例题精讲
例 1 计算:
1
1×2
+
1
2×3
+
1
3×4
+ … +
1 49×50
解 原式 =( 1
1
-
1
2
) +(
1
2
1
3
) +(
1
3
1
4
) +… +(
1
49
1 50 )
= 1
1
-
1
2
1
2
-
1
3
1
3
-
1
4
… +
1 49 1 50
=1- 1 50 49 50
例 2 计算 :
1
2×4
+
1
4×6
+
1
6×8
… +
1 98×100
解 原式 = 1
2
(
1
2
1
4
)+
1
2
1
4
1
6
)+
1
2
1
6
-
1
8
… +
1
2
1
98
- 1 100
= 1
2
(
1
2
1
4
+
1
4
1
6
+
1
6
1
8
… +
1
98
1 100
= 1
2
(
1
2
1
100
)
= 1
2
×
49 100 = 49 200
例 3 计算: 1
1×2×3
1
2×3×4
+ … +
1 98×99×100
解 因为
1
1×2×3
1
2
×(
1
1×2
-
1
2×3
),
1
2×3×4
1
2
×(
1
2×3
-1 3×4
1
98×99×100 =
1
2
×(
1
98×99
1 99×100
所以 原式 = 1
2
1
1×2
-
1
2×3
+
1
2×3
-
1
3×4
… +
1
98×99
-
1
99×100 )
= 1
2
1
2
-
1
99×100
) =
4949 19 800
例 4计算 : 1+
1
1+2
1
1+2+3
1
1+2+3+4
… +
1
1+2+3+… +99+100
解 原式 =1+
1
(1+4) ×4
2
+
1
(1+3) ×3
2
+
1
(1+4) ×4
2
… +
1 (1+100)×100 2
= 1+
2
2×3
2
3×4
2
4×5
+… +
2 100×101
= 2×( 1
1×2
1
2×3
1
3×4
1
4×5
+… +
1 100×101
= 2×(1- 1
2
1
2
-
1
3
1
3
-
1
4
1
4
-
1
5
… +
1
100
1 101 )
= 2×(1-
1 101 )
= 1 99 101
例 5 计算 : 1
2
1
2+3
1
2+3+4
+
1
2+3+4+5
… +
1
2+3+4+… +199+200
解 原式 = 1
2
+
1
(2+3) ×2
2
1
(2+4) ×3
2
+
1
(2+5) ×4
2
… +
1
(2+200) ×199 2
= 1
2
+
2
2×5
2
3×6
2
4×7
… +
2 199×202
= 1
2
+ 2×(
1
2×5
+
1
3×6
+
1
4×7
… +
1 199×202
= 1
2
+
2
3
×(
1
2
-
1
5
1
3
-
1
6
1
4
1
7
+… +
1
199
-
1 202 )
= 1
2
+
2
3
×[(
1
2
+
1
3
1
4
+… +
1
199
)-(
1
5
1
6
+
1
7
… + 1 202
= 1
2
+
2
3
×(
1
2
+
1
3
1
4
-
1
200
1
201
-
1 202 )
= 1
2
+
2
3
×(
99
200
+
6
201
+ 99 404
= 1
430 933 2 030 100
范文二:分数数列求和
分数数列求和
例题精讲
1111 例 1 计算: + + + … + 1×22×33×449×50
11111111 解 原式=( - )+( - )+( + )+…+( - ) 122334495011111111 = - + - + - +…+ - 1223344950
149 =1- = 5050
1111例 2 计算: + + +…+ 2×44×66×898×100
111111111111 解 原式= ( - )+ ( - )+ ( - )+…+ ( - ) 224246268298100111111111 = ( - + - + - +…+ - ) 224466898100
111 = ( - ) 22100
14949 = × = 2100200
111例3 计算: + + …+ 1×2×32×3×498×99×100
11111111解 因为 = ×( - ), = ×( - ), 1×2×321×22×32×3×422×33×4
1111 = ×( - ) 98×99×100298×9999×100
1111111 所以 原式= ( - + - +…+ - ) 21×22×32×33×498×9999×100
1114949 = ( - ) = 2299×10019 800
1111例 4计算: 1+ + + +…+ 1+21+2+31+2+3+41+2+3+…+99+100
1111解 原式=1+ + + +…+ 1+4) ×41+3) ×31+4) ×41+100)×100(((( 2222
2222 = 1+ + + +…+ 2×33×44×5100×101
11111 = 2×( + + + +…+ ) 1×22×33×44×5100×101
111111111 = 2×(1- + - + - + - +…+ - ) 2233445100101
1- ) = 2×(1101
99 = 1 101
11111例 5 计算: + + + +…+ 22+32+3+42+3+4+52+3+4+…+199+20011111 解 原式= + + + +…+ 2(2+3) ×2(2+4) ×3(2+5) ×4(2+200) ×199 222212222 = + + + +…+ 22×53×64×7199×202
11111 = + 2×( + + +…+ ) 22×53×64×7199×2021211111111 = + ×( - + - + - +…+ - ) 232536471992021211111111 = + ×[( + + +…+ )-( + + +…+ )] 2323419956720212111111 = + ×( + + - - - ) 23234200201202
1299699 = + ×( + + ) 23200201404
430 933 = 1 2 030 100
范文三:分数数列求和YN
六年级
分数数列求和 【一星级题】
【二星级题】
11111111,,,,,,,, 1( 2( 1223344950,,,,24466898100,,,,
3333333,,,,,, 3( 14477101013131616191922,,,,,,,
222226810202224,,,,,,,,,,, 4( 5( 2177165167720217911212325,,,,,,
199519951995199519951995,,,,,, 6( 122334455619941995,,,,,,
1111,,,, 7( 13355719971999,,,,
111111,,,,, 8( 2558811111414171720,,,,,,
111111,,,,,, 9( 2771212171722929797102,,,,,,
11111111111111,,,,,,,,,,,, 10( 11( 304256729011013212203042567290
11111,,,,, 12( 12233419202021,,,,,
11111,,,,, 13( 1447710101397100,,,,,
111112222,,,,,,,, 14( 15( 31535639911131315151719951997,,,,
111112222,,,,,,,,, 16( 17( 12233489910,,,,,13355799101,,,,
1
六年级
111111111,,,,,,,,, 18( 19( 315353994831661111169196,,,,
11111111,,,,,,,, 20( 21( 14477103134,,,,13243519992001,,,,
12320022003,,,,, 22. 20032003200320032003
11111,,,,, 23( 1223344520042005,,,,,
111111,,,,,, 24( 1335577999101,,,,,
111111111,,,,,,,,, 25( 26( 2612990027712121797102,,,,
111111111,,,,,,,,, 27( 28( 36101545104088928
【三星级题】
11111111111315171921232527,,,,,,,, 1( 612203042567290
15111095791113151719,,,,1,,,,,,,, 2( 3( 2612110612203042567290
11212312341235859,,,,,,,,,,,,,,,,()()()() 4( 23344455556060606060
199219911990198919881987198619854321,,,,,,,,,,,, 5( 199319931993199319931993199319931993199319931993
457891111111,,,,()64,,,,,,, 6( 7( 361215208244880120168224
1111111111,,,,,3579111246,,, 8( 9( 15356399173195104088154
2
六年级
1111113579,,,,,,,,, 10( 104088154238
111111111111357911,,,,,1357910,,,,,, 11( 12( 612203042315356399143
111111111357911131517,,,,,,,, 13( 612203042567290
57911131517191,,,,,,,, 14( 612203042567290
12310111220,,,,,,,, 15( 55555555155155155
16( (1.00.1)(2.00.2)(3.00.3)(9.00.9)(10.01.0),,,,,,,,,
111111111357911131517,,,,,,,, 17( 612203042567290
1111,,,, 18( 123234345181920,,,,,,,,
11111,,,,,,,,,,,,,1248128256512 19( 51225612842
1111,,, 20( 456567678789,,,,,,,,
2222,,,, 21( 2464686810161820,,,,,,,,
111,,, 22( 1232349899100,,,,,,
1111111,,,,,, 23( 81282545081016203240648128
111111,,,,,357911 24( 2558811111414171720,,,,,,
3
六年级
15111989,,,,, 25( 26122090
11111,,,,, 26( 12323434545691011,,,,,,,,,,
1111,,,, 27( 10111213111213141213141517181920,,,,,,,,,,,,
57911131517191,,,,,,,, 28( 612203042567290
11111511198912398,,,,135717,,,,, 29( 30( 1223349899,,,,26122090
33333,,,,, 31( 345456567678181920,,,,,,,,,,
12348,,,,, 32( 1232343454568910,,,,,,,,,,
1111,,,, 33( 123452345634567678910,,,,,,,,,,,,,,,,
17911131517157911511,,,,,,,,,,,,, 34( 35( 3122030425672261220305642
【四星级题】
222221,,,,, 1( 22222221416181101121,,,,,,
22222224681012 2( ,,,,,133557799111113,,,,,,
1111111119941234519921993,,,,,,,,, 3( 23232323
11211232111,,,,,,,,,,,, 4( 12223333319921992
4
六年级
111213110(1)(2)(3)(10),,,,,,,, 5( 23232323
112112321123321 6(,,,,,,,,,,,,,,,, 122233333100100100100100100
1111111119921234519901991,,,,,,,,, 7( 23232323
11111,,,,, 8( 121231234123100,,,,,,,,,,
1111,,,, 9( 224246246100,,,,,,,
234 10(和式 ,,,,112121231231234,,,,,,,,,,,,,,,,,,,,,,
100 计算化简后得到一个最简分数,其分母与分123989912399100,,,,,,,,,,,,,,,
子之差为多少,
11111948 11(的最小自然数n为 。 ,,,,,12233411998,,,,,nn,,
234 12( 1,,,,,112121231231234,,,,,,,,,,,,,,,,,,,,,,
10 1239123910,,,,,,,,,,,,,,
111,,,,,,1223341112,,,,,,,, 13( 14(111,,,,,, ,,,,,,2222399,,,,,,
11111,,,,, 15( 121231234123499100,,,,,,,,,,,,
5
六年级
1111 16(,,,, 23234234523499100,,,,,,,,,,,
1111,,,, 17( 223234234199200,,,,,,,,
234 18( ,,,,112121231231234,,,,,,,,,,,,,,,,,,,,,,
2004 1232003122004,,,,,,,,,,,,
11111,,,,, 19( 1121231234123499100,,,,,,,,,,,,
【五星级题】
1212312341250,,,,,,,,,,,,, 1( 2232342350,,,,,,
1236,,,, 2( 223234237,,,,,,
123456,,,,, 3( 223234234523456234567,,,,,,,,,,,,,,,
12348,,,,, 4( 223234234523456789,,,,,,,,,,,,,
6
范文四:分数数列求和
六年级
分数数列求和 【一星级题】
【二星级题】
11111111 1( 2( ,,,,,,,,1223344950,,,,24466898100,,,,
3333333 3( ,,,,,,14477101013131616191922,,,,,,,
222226810202224,,,,,, 4( 5( ,,,,,2177165167720217911212325,,,,,,
199519951995199519951995 6( ,,,,,,122334455619941995,,,,,,
1111 7( ,,,,13355719971999,,,,
111111 8( ,,,,,2558811111414171720,,,,,,
111111 9( ,,,,,,2771212171722929797102,,,,,,
11111111111111 10( 11( ,,,,,,,,,,,,304256729011013212203042567290
11111 12( ,,,,,12233419202021,,,,,
11111 13( ,,,,,1447710101397100,,,,,
111112222 14( 15( ,,,,,,,,31535639911131315151719951997,,,,
111112222 16( 17( ,,,,,,,,,12233489910,,,,,13355799101,,,,
1
六年级
111111111 18( 19( ,,,,,,,,,315353994831661111169196,,,,
11111111 20( 21( ,,,,,,,,14477103134,,,,13243519992001,,,,
12320022003 22. ,,,,,20032003200320032003
11111 23( ,,,,,1223344520042005,,,,,
11111 24( 1,,,,,,1335577999101,,,,,
11111111 25( 26( 1,,,,,,,,,2612990027712121797102,,,,
111111111 27( 28( ,,,,,,,,,36101545104088928
【三星级题】
11111111 1( 111315171921232527,,,,,,,,612203042567290
15111095791113151719 2( 3( ,,,,1,,,,,,,,2612110612203042567290
11212312341235859 4( ,,,,,,,,,,,,,,,,()()()()23344455556060606060
199219911990198919881987198619854321 5( ,,,,,,,,,,,,199319931993199319931993199319931993199319931993
457891111111 6( 7( ,,,,()64,,,,,,,361215208244880120168224
1111111111 8( 9( ,,,,,3579111246,,,15356399173195104088154
2
六年级
11111 10( 13579,,,,,,,,,104088154238
11111111111 11( 12( 1357911,,,,,1357910,,,,,,612203042315356399143
11111111 13( 1357911131517,,,,,,,,612203042567290
5791113151719 14( 1,,,,,,,,612203042567290
12310111220 15( ,,,,,,,,55555555155155155
16( (1.00.1)(2.00.2)(3.00.3)(9.00.9)(10.01.0),,,,,,,,,
11111111 17( 1357911131517,,,,,,,,612203042567290
1111 18( ,,,,123234345181920,,,,,,,,
11111 19( ,,,,,,,,,,,,,124812825651251225612842
1111 20( ,,,456567678789,,,,,,,,
2222 21( ,,,,2464686810161820,,,,,,,,
111 22( ,,,1232349899100,,,,,,
1111111 23( ,,,,,,81282545081016203240648128
111111 24( ,,,,,3579112558811111414171720,,,,,,
3
六年级
15111989 25( ,,,,,26122090
11111 26( ,,,,,12323434545691011,,,,,,,,,,
1111 27( ,,,,10111213111213141213141517181920,,,,,,,,,,,,
5791113151719 28( 1,,,,,,,,612203042567290
111115111989 29( 30( 12398,,,,135717,,,,,1223349899,,,,26122090
33333 31( ,,,,,345456567678181920,,,,,,,,,,
12348 32( ,,,,,1232343454568910,,,,,,,,,,
1111 33( ,,,,123452345634567678910,,,,,,,,,,,,,,,,
1791113151715791151 34( 35( 1,,,,,,,,,,,,,3122030425672261220305642
【四星级题】
222221 1( ,,,,,22222221416181101121,,,,,,
22222224681012,,,,, 2( 133557799111113,,,,,,
11111111 3( 19941234519921993,,,,,,,,,23232323
11211232111 4( ,,,,,,,,,,,,12223333319921992
4
六年级
111213110 5( (1)(2)(3)(10),,,,,,,,23232323
112112321123321 6( ,,,,,,,,,,,,,,,,122233333100100100100100100
11111111 7( 19921234519901991,,,,,,,,,23232323
1111 8( 1,,,,,121231234123100,,,,,,,,,,
1111 9( ,,,,224246246100,,,,,,,
234,,,, 10(和式 112121231231234,,,,,,,,,,,,,,,,,,,,,,
100 计算化简后得到一个最简分数,其分母与分123989912399100,,,,,,,,,,,,,,,
子之差为多少,
11111948,,,,, 11(的最小自然数n为 。 12233411998,,,,,nn,,
2341,,,,, 12( 112121231231234,,,,,,,,,,,,,,,,,,,,,,
10 1239123910,,,,,,,,,,,,,,
111,,,,,,111,,,,,, 13( 14( 1223341112,,,,,,,,,,,,,,2222399,,,,,,
1111 15( 1,,,,,121231234123499100,,,,,,,,,,,,
5
六年级
1111 16( ,,,,23234234523499100,,,,,,,,,,,
1111 17( ,,,,223234234199200,,,,,,,,
234 18(,,,, 112121231231234,,,,,,,,,,,,,,,,,,,,,,
2004 1232003122004,,,,,,,,,,,,
11111 19( ,,,,,1121231234123499100,,,,,,,,,,,,
【五星级题】
1212312341250,,,,,,,,, 1( ,,,,2232342350,,,,,,
1236 2( ,,,,223234237,,,,,,
123456 3( ,,,,,223234234523456234567,,,,,,,,,,,,,,,
12348 4( ,,,,,223234234523456789,,,,,,,,,,,,,
6
范文五:分数数列求和2
11111、 + + + … + 1×22×33×449×50
111,,??,2、 10,1111,1259,60
11111111,,,,,,,3、 612203042567290
11111,,,,?,4、? 1,21,2,31,2,3,41,2,3,?,99,100
111111111,2,3,4,5,6,7,8,95、 612203042567290
1791113151,,,,,6、 31220304256
111111、,,,, 5,77,99,1111,1313,15
1111,,,?,2、 1,44,77,1097,100
11111,3,5,9,113、 2,55,88,1111,1414,17
11111,,,,4、 315356399
11111,,,,?,5、 2,53,64,75,87,10
11111,,,,?,6、 1,32,43,54,620,22
1111,,,?,7、 22,42,4,62,4,6,?,100