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模块二
数的运算
专题6
四则混合运算
类型一 直接计算
1.
解:原式=(3. 2+0. 75-3. 2)+1. 8
= 0. 75+1. 8
= 2. 55
2.
解:原式=(6. 4-5. 2)× 7
6 +3. 3× 6
11
= 1. 2× 7
6 +1. 8
= 3. 2
3.
解:原式=( 5
4 +35
12×16
7 )× 4
15
=( 5
4 +20
3 )× 4
15
= 95
12× 4
15
= 19
9
4.
解:原式=(3. 85×18
5 +27. 675)× 4
13-1 1
13
=(13. 86+27. 675)× 4
13-1 1
13
= 41. 535× 4
13-1 1
13
= 8307
650 -700
650
= 7607
650
5.
解:原式=[16-(2. 7+24
25× 5
16)×19
12]×8
=[16-(2. 7+0. 3)×19
12]×8
=(16-19
4 )×8
= 16×8-19
4 ×8
= 128-38
= 90
6.
解:原式= 7
2 × 3
4 +13
2 ×[12×( 4
3 - 3
10)- 3
20]
= 21
8 +13
2 ×(12×31
30- 3
20)
= 21
8 +13
2 ×49
4
= 21
8 +637
8
= 329
4
7.
解:原式= 1
80×16
5 + 1
7 ×175
2 × 4
15× 4
15+4-4
= 1
25+ 8
9
= 209
225
8.
解:原式= 1 2
9 ×1 8
33+19
27
= 11
9 ×41
33+19
27
= 41
27+19
27
= 20
9
9.
解:原式= 2012×
15
4 ×13
10+3× 3
8
25×20+3
= 2012×
39
8 + 9
8
503
= 2012× 6
503
= 24
类型二 繁分数(多层分数线叠加)
10.
解:原式=
1
1+
1
2+ 1
13
4
=
1
1+
1
2+ 4
13
=
1
1+ 1
30
13
=
1
1+13
30
= 1
43
30
= 30
43
11.
解:原式=
11
3
6 + 2
6 + 1
6
-
3
6 + 2
6 + 1
6
11
2×3×6
=
11- 1
11
2×3×6
=
120
11
2×3×6
= 120
11 ×
1
2×3×6
= 10
33
12.
解:原式=
(128
45 - 3
45)÷125
9 +198
65 ×26
99
( 37
2 -124
9 )× 1
85
=
125
45 × 9
125+198
65 ×26
99
85
18× 1
85
=
1
5 + 4
5
1
18
= 18
13.
解:原式=
(- 1
6 )
(-51) +
(-4 1
4 )+4+ 1
4
2
2023
=
(- 1
6 )
(-51) + 0
2
2023
= 1
6 × 1
51
= 1
306
类型三 列式计算
14.
9 【解析】66÷(7+ 1
3 )= 9。
15.
100 【解析】35÷(60%-25%)= 100。
16.
解:(120×20%+24)÷ 4
5 = 60。
答:这个数是60。
类型四 使算式成立
17.
2×7+1+9= 24(答案不唯一)
18.
407 【解析】得数最大的添法为1+ 2×(3+ 4× 5+ 6) × 7
= 407。
19.
(1)6×(24÷3-5)+12= 30 (2)(9+4)×(14+63)÷7= 143
20.
12+3+4+56+7+8+9= 99(答案不唯一)
专题7
四则运算律和性质
类型一 加减法的运算律和性质
1.
解:原式= 1+(22+8)+(333+77)+(4444+666)+5555+1
= 1+30+410+5110+5555+1
= 11107
2.
解:原式=(8 3
4 +16 1
4 +1 1
4 -3 1
4 )-(5 4
11+6 7
11)
= 23-12
= 11
3.
解:原式=
19
4 × 8
19+44
3 ×10
11
(10. 68+4. 32)-(0. 36+1. 64)
=
2+40
3
13
= 46
39
4.
解:原式= 17-( 13
6 -2. 5+11
6 )÷0. 125
= 17-1. 5÷0. 125
= 17-12
= 5
5.
解:原式=(4036 2
3 -2017 2
3 )-(2018 5
6 +1 1
6 )
= 2019-2020
= -1
6.
解:原式= 13 8
11-3 1
4 -2 5
11- 3
4
= 13 8
11-2 5
11-(3 1
4 + 3
4 )
= 11 3
11-4
= 7 3
11
类型二 乘除法的运算律和性质
7.
解:原式= 18×(4×0. 25)
= 18×1
= 18
8.
解:原式=(241÷241)×(690÷345)×(678÷339)
= 1×2×2
= 4
9.
解:原式= 1. 25×8×4×2. 5
=(1. 25×8)×(4×2. 5)
= 10×10
= 100
1
10.
解:原式= 0. 25×4×8×4×0. 125× 3
4
=(0. 25×4)×(8×0. 125)×(4× 3
4 )
= 1×1×3
= 3
11.
解:原式= 0. 6×3. 7+0. 6+5. 3×0. 6
= 0. 6×(3. 7+1+5. 3)
= 6
12.
解:原式= 3
2022×(3
2-5×3+6)
= 3
2022×0
= 0
13.
解:原式= 17. 4×37-17. 4×19+17. 4×82
= 17. 4×(37-19+82)
= 17. 4×100
= 1740
14.
解:原式= 3. 14×4. 3+3. 14×7. 2-3. 14×1. 5
= 3. 14×(4. 3+7. 2-1. 5)
= 3. 14×10
= 31. 4
15.
解:原式= 7
2 × 5
4 +1× 5
4 + 3
2 × 5
4
=( 7
2 +1+ 3
2 )× 5
4
= 6× 5
4
= 15
2
16.
解:原式= 48× 23
135+37× 23
135+25×2× 23
135
= 23
135×(48+37+50)
= 23
135×135
= 23
17.
解:原式= 9
5 × 3
4 +0. 23×0. 75+0. 37×0. 75
= 1. 8×0. 75+0. 23×0. 75+0. 37×0. 75
= 0. 75×(1. 8+0. 23+0. 37)
= 0. 75×2. 4
= 1. 8
18.
解:原式=(724+1)×876-576
300+876×724
+(88+12)×65+(99+11)×60
= 724×876+876-576
300+876×724
+100×65+110×60
= 724×876+300
300+876×724+6500+6600
= 1+6500+6600
= 13101
19.
解:原式= 83×111. 1+111. 1×9×13
= 111. 1×(83+9×13)
= 111. 1×200
= 22220
20.
解:原式= 38 3
4 ×790+790×61 1
4
=(38 3
4 +61 1
4 )×790
= 100×790
= 79000
21.
解:原式= 33. 4×6. 5+3. 34×3. 95÷0. 1-33. 4×0. 45
= 33. 4×(6. 5+3. 95-0. 45)
= 334
22.
解:原式= 314×0. 314+628×0. 686+686×0. 686
= 314×0. 314+(628+686)×0. 686
= 314×0. 314+1314×0. 686
= 314×0. 314+(314+1000)×0. 686
= 314×0. 314+314×0. 686+686
= 314×(0. 314+0. 686)+686
= 1000
23.
解:原式=(1+ 1
2 + 1
4 )×( 1
2 + 1
4 + 1
6 )-(1+ 1
2 + 1
4 )×( 1
2 +
1
4 )- 1
6 ×( 1
2 + 1
4 )
= ( 1 + 1
2 + 1
4 ) ×
1
2 + 1
4 + 1
6 -( 1
2 + 1
4 )
é
ëêê
ù
ûúú - 1
6 ×
( 1
2 + 1
4 )
=(1+ 1
2 + 1
4 )× 1
6 - 1
6 ×( 1
2 + 1
4 )
= 1
6 ×(1+ 1
2 + 1
4 - 1
2 - 1
4 )
= 1
6 ×1
= 1
6
24.
解:原式= 94
3 ×83
2 -34
3 ×83
2 ×2-19
2 ×34
3
= 1
6 ×(94×83-34×83×2-19×34)
= 1
6 ×(94×83-34×83-34×83-19×34)
= 1
6 ×[83×(94-34)-34×(83+19)]
= 1
6 ×(83×60-34×102)
= 83×10-34×17
= 252
25.
解:原式= 2013×[(4. 8-0. 1)×73+3. 6]÷(4. 8×73-3. 7)
= 2013×(4. 8×73-0. 1×73+3. 6)÷(4. 8×73-3. 7)
= 2013×[4. 8×73-(7. 3-3. 6)]÷(4. 8×73-3. 7)
= 2013×(4. 8×73-3. 7)÷(4. 8×73-3. 7)
= 2013
26.
解:原式= 9
8 ×7 7
13×8+8×1 1
13× 9
8 - 9
8 ×1 3
13×8+4 12
13× 9
8
= 9×7 7
13+9×1 1
13-9×1 3
13+9× 8
13
= 9×(7 7
13+1 1
13-1 3
13+ 8
13)
= 9×8
= 72
27.
解:原式=(11+12+…+21)- 4
66×(1+2+…+11)
= 176- 4
66×66
= 172
28.
解:原式=(4433-443. 3)÷(4433×2-443. 3×2)
=(4433-443. 3)÷[(4433-443. 3)×2]
=(4433-443. 3)÷(4433-443. 3)÷2
= 1÷2
= 1
2
29.
解:原式= 0. 84× 3
20×0. 45÷0. 54÷0. 14÷ 3
10
=(0. 84÷0. 14)×( 3
20÷ 3
10)×(0. 45÷0. 54)
= 6× 1
2 × 5
6
= 5
2
30.
解:原式= 13. 5×(48-9. 6)÷4. 8
= 13. 5×(48÷4. 8-9. 6÷4. 8)
= 13. 5×(10-2)
= 13. 5×8
= 108
专题8
变形约分
类型一 变形后逐个约分
1.
解:原式= 52
51×50
51×53
52×51
52×…×153
152×151
152
= 50
51×153
152
= 75
76
2.
解:原式= 522÷522×523+522×1
523
+526÷(526+1)×523+3
523
=522×
523
522×(523+1)+526×
523
526×523+(523+3)
= 523
524+526×
523
526×(523+1)
= 523
524+523
524
= 523
262
3.
解:原式=(57×85×105×117×121)÷[1×(3×19)×(5×17) ×
(7×15)×(9×13)×11]
=(57×85×105×117×121)÷(57×85×105×117×11)
= 121÷11
= 11
4.
解:原式= 1×3+1
1×3 ×2×4+1
2×4 ×3×5+1
3×5 ×…×2023×2025+1
2023×2025
= 2
2
1×3× 3
2
2×4× 4
2
3×5×…×
2024
2
2023×2025
= 2
1 × 2
3 × 3
2 × 3
4 × 4
3 × 4
5 ×…×2024
2023×2024
2025
= 2
1 ×2024
2025
= 4048
2025
5.
解:原式= 2
2-1
2
2 ×3
2-1
3
2 ×4
2-1
4
2 ×…×9
2-1
9
2 ×10
2-1
10
2
= 3×1
2×2×4×2
3×3×5×3
4×4×…×10×8
9×9 × 11×9
10×10
= 1
2 ×11
10
= 11
20
类型二 变形后整体约分
6.
解:原式= 411×362+(362+1)
411×(362+1)-48
= 362×(411+1)+1
411×362+411-48
= 362×411+363
411×362+363
= 1
7.
解:原式=
2010×2009+1
2009
2009×2010+1
2010
+
2008×2009+1
2009
2009×2008+1
2008
= 2010
2009+2008
2009
= 4018
2009
2
= 2
8.
解:原式= 2012+2014×(2012-1)
2012×2014-2
+
65
7 +65
9
5
7 + 5
9
= 2012×2014-2
2012×2014-2+
65×( 1
7 + 1
9 )
5×( 1
7 + 1
9 )
= 1+13
= 14
9.
解:原式=
(3+ 6
7 )-(3+ 6
8 )+(3+ 6
9 )-(3+ 6
10)
1
7 - 1
8 + 1
9 - 1
10
=
6
7 - 6
8 + 6
9 - 6
10
1
7 - 1
8 + 1
9 - 1
10
=
6×( 1
7 - 1
8 + 1
9 - 1
10)
1
7 - 1
8 + 1
9 - 1
10
= 6
10.
解:原式= 2×3×4×(1
3+2
3+3
3+…+15
3)
1×2×3×(1
3+2
3+3
3+…+15
3)
= 2×3×4
1×2×3
= 4
11.
解:原式=
1×3×9×[1. 2
3+2
3+( 1
13)
3
]
1×2×4×[1. 2
3+2
3+( 1
13)
3
]
= 1×3×9
1×2×4
= 27
8
12.
解:原式=(
5 1
3 +2 1
4 -1 1
5
1 1
3 +1 1
2
+1)×
1 1
3 +1 1
2
6 2
3 +1 2
4 +2 1
4 -1 1
5
=
5 1
3 +2 1
4 -1 1
5 +1 1
3 +1 1
2
1 1
3 +1 1
2
×
1 1
3 +1 1
2
6 2
3 +3 3
4 -1 1
5
=
6 2
3 +3 3
4 -1 1
5
6 2
3 +3 3
4 -1 1
5
= 1
13.
解:原式=
8
1 × 9
2 ×10
3 ×11
4 ×12
5 ×13
6 ×14
7 ×15
8 ×16
9
10
1 ×11
2 ×12
3 ×13
4 ×14
5 ×15
6 ×16
7
=
8×9×10×11×12×13×14×15×16
1×2×3×4×5×6×7×8×9
10×11×12×13×14×15×16
1×2×3×4×5×6×7
= 1
14.
解:原式=
( 4
7 ×10
9 × 4
11)÷( 2
11× 2
7 × 5
9 )
( 4
7 +
10
9 +
4
11)÷( 2
11+
2
7 +
5
9 )
=
(2× 2
7 ×2× 5
9 ×2× 2
11)÷( 2
11× 2
7 × 5
9 )
(2× 2
7 +2× 5
9 +2× 2
11)÷( 2
11+
2
7 +
5
9 )
=
(2×2×2)×[( 2
7 × 5
9 × 2
11)÷( 2
11× 2
7 × 5
9 )]
2×[( 2
7 +
5
9 +
2
11)÷( 2
11+
2
7 +
5
9 )]
= 2×2×2
2
= 4
15.
解:原式=
1
2+2
2+3
2+…+19
2+20
2
2
2×1
2+2
2×2
2+2
2×3
2+…+2
2×19
2+2
2×20
2
=
1
2+2
2+3
2+…+19
2+20
2
2
2×(1
2+2
2+3
2+…+19
2+20
2)
= 1
4
类型三 应用“重码数”特点约分
16.
解:原式= 2024×2025×10001-2025×2024×10001
= 0
17.
解:原式= 3×111×7×111-2×111×6×111
5×111×9×111
= 3×7-2×6
5×9
= 1
5
18.
解:原式=
111111×111111
7×111111×9×111111
= 1
63
19.
解:原式=( 1
31+ 2×101
31×101+ 5×10101
31×10101+13×1010101
31×1010101)÷3
=( 1
31+ 2
31+ 5
31+13
31)÷3
= 21
31÷3
= 7
31
20.
解:原式= 333×(332332332+1)-(333-1)×333333332
= 333×332332332+333-333×333333332
+333333332
= 333+333333332-333×(333333332-332332332)
= 333333665-333×1001000
= 333333665-333333000
= 665
21.
解:原式=(56×101)×(13×101)+(21×101)×(26×101)
(39×101)×(28×101)
= 101
2×56×13+101
2×21×26
101
2×39×28
= 7×8×13+7×3×13×2
13×3×7×4
=(7×13)×(8+3×2)
(13×7)×(3×4)
= 7
6
22.
解:原式=(191919
989898+190190
980980+19001900
98009800)×98
19×9898
1919
=(191919
989898×98
19+190190
980980×98
19+19001900
98009800×98
19)×9898
1919
=(10101
10101+10010
10010+1000100
1000100)×9898
1919
= 3×9898
1919
= 294
19
23.
解: 原式= 31×11
25×11 + 2 × 31×111
25×111 + 3 × 31×1111
25×1111 + … + 8 ×
31×111111111
25×111111111+9×31×1111111111
25×1111111111
= 31
25+2×31
25+3×31
25+…+8×31
25+9×31
25
= 31
25×(1+2+3+4+5+6+7+8+9)
= 279
5
专题9 灵活拆分与分组、裂项相消
类型一 拆分与分组
1.
解:原式=(999. 3-998. 2)+(997. 3-996. 2)+…+(3. 3-2. 2)+
(1. 3-0. 2)
= 1. 1+1. 1+1. 1+…+1. 1
= 1. 1×500
= 550
2.
解:原式= 1
2 +( 1
3 + 2
3 ) +( 1
4 + 2
4 + 3
4 ) +…+( 1
9 + 2
9 +…+
7
9 + 8
9 )
= 1
2 + 2
2 + 3
2 +…+ 8
2
= 1
2 ×(1+2+3+…+8)
= 1
2 ×36
= 18
3.
解:原式= 1+( 1
2 + 2
2 + 1
2 ) +( 1
3 + 2
3 + 3
3 + 2
3 + 1
3 ) +
…+
(
1
2010+ 2
2010+…+2010
2010+…+ 2
2010+ 1
2010)
= 1+2+3+…+2010
=(1+2010)×2010÷2
= 2021055
4.
解:分子=(1- 1
2 - 1
4 - 1
8 - 1
16) +( 1
3 - 1
6 - 1
12) +( 1
5 - 1
10 -
1
20)+( 1
7 - 1
14)+( 1
9 - 1
18)+ 1
11+ 1
13+ 1
15+ 1
17+ 1
19
= 1
11+ 1
12+ 1
13+…+ 1
19+ 1
20
分母= 1
22+ 1
24+…+ 1
40 = 1
2 ×( 1
11+ 1
12+ 1
13+…+ 1
19+ 1
20)
原式=
1
11
+
1
12
+
1
13
+
…
+
1
19
+
1
20
1
2 ×( 1
11
+
1
12
+
1
13
+
…
+
1
19
+
1
20)
= 2
5.
解:原式= 1111
1399×(1399+1)
= 1111
1399×1399+1111
1399×1
= 1111+1111
1399
= 1111 1111
1399
6.
解:原式= 2012×
15
4 ×1. 3+3× 3
8
25×20+3
= 2012×
( 15
4 ÷10)×(1. 3×10)+3× 3
8
503
= 2012×
3
8 ×13+3× 3
8
503
3
= 2012×
3
8 ×(13+3)
503
= 2012× 6
503
= 24
7.
解:原式= 999 8
9 +99 8
9 +9 8
9 +( 1
9 ×3)
= 999+ 8
9 +99+ 8
9 +9+ 8
9 + 1
9 + 1
9 + 1
9
=(999+ 8
9 + 1
9 )+(99+ 8
9 + 1
9 )+(9+ 8
9 + 1
9 )
= 1000+100+10
= 1110
8.
解:原式=(1+2+3+4+5+6)- 1
21×(1+2+3+4+5+6)
= 21- 1
21×21
= 21-1
= 20
9.
解: 原式= (2012-1)×(2012+1)
2012
+ (2013-1)×(2013+1)
2013
+
2012+2013
2012×2013
= 2012
2
2012 - 1
2012+2013
2
2013 - 1
2013+ 1
2012+ 1
2013
= 2012- 1
2012+2013- 1
2013+ 1
2012+ 1
2013
= 2012+2013
= 4025
10.
解:原式=(1×100000+2×10000+3×1000+4×100+5×10+
6×1)+(2×100000+3×10000+4×1000+5×100+6×
10+1×1)+…+(6×100000+1×10000+2×1000+3×
100+4×10+5×1)
= 111111+222222+333333+444444+555555+666666
=(1+2+3+4+5+6)×111111
= 21×111111
=(20+1)×111111
= 2222220+111111
= 2333331
类型二 拆分与裂项相消
11.
解:原式= 1- 1
2 + 1
2 - 1
3 + 1
3 - 1
4 …+ 1
2025- 1
2026
= 1- 1
2026
= 2025
2026
12.
解:原式= ( 1
11 - 1
13) +( 1
13 - 1
15) +( 1
15 - 1
17) +( 1
17 - 1
19) +
( 1
19- 1
21)+ 1
21
= 1
11
13.
解:原式=(6+12+20+30+42+56+72) +( 1
3 + 1
6 + 1
10+ 1
15+
1
21+ 1
28+ 1
36)
=238+[ 1
3 + 1
6 + 1
3 ×( 1
2 - 1
5 )+ 1
15+ 1
4 ×( 1
3 - 1
7 )+
1
28+ 1
36]
= 238+( 1
3 + 1
6 + 1
6 - 1
15+ 1
15+ 1
12- 1
28+ 1
28+ 1
36)
= 238+( 1
3 + 1
6 + 1
6 + 1
12+ 1
36)
= 238 7
9
14.
解:原式= ( 1 + 1
1×3) + ( 1 + 1
3×5) + ( 1 + 1
5×7) + … + ( 1 +
1
99×101)
= 1×50+ 1
2 ×(1- 1
3 + 1
3 - 1
5 + 1
5 - 1
7 +…+ 1
99- 1
101)
= 50+ 1
2 ×(1- 1
101)
= 50+ 1
2 ×100
101
= 50 50
101
15.
解:原式= 1
30×( 1
3 - 1
33)+ 1
30 ×( 1
33 - 1
333)+ 1
30 ×( 1
333- 1
3333)+
1
30×( 1
3333-
1
33333)
= 1
30×( 1
3 - 1
33+ 1
33- 1
333+ 1
333- 1
3333+ 1
3333-
1
33333)
= 1
30×( 1
3 -
1
33333)
= 1
30×11110
33333
= 1111
99999
16.
解:原式=(1+ 1
2 )-(3- 1
6 )+(3+ 1
12)-(5- 1
20)+(5+ 1
30)-
(7- 1
42)+(7+ 1
56)-(9- 1
72)+(9+ 1
90)
= 1+ 1
2 + 1
6 + 1
12+ 1
20+ 1
30+ 1
42+ 1
56+ 1
72+ 1
90
= 1+(1- 1
2 ) +( 1
2 - 1
3 ) +( 1
3 - 1
4 ) +( 1
4 - 1
5 ) +
( 1
5 - 1
6 )+( 1
6 - 1
7 )+( 1
7 - 1
8 )+( 1
8 - 1
9 )+( 1
9 -
1
10)
= 2- 1
10
= 19
10
17.
解:原式=
1
2
3
2
+
1
3
3
2 × 4
3
+
1
4
3
2 × 4
3 × 5
4
+…+
1
99
3
2 × 4
3 ×…×100
99
= 2
2×3+ 2
3×4+ 2
4×5+…+
2
99×100
= 2×( 1
2×3+ 1
3×4+ 1
4×5+…+
1
99×100)
= 2×( 1
2 - 1
3 + 1
3 - 1
4 + 1
4 - 1
5 +…+ 1
99- 1
100)
= 2×( 1
2 - 1
100)
= 49
50
18.
解:原式= ( 1
2 - 1
7 ) +( 1
7 - 1
12) +( 1
12 - 1
17) +( 1
17 - 1
22) +
( 1
22- 1
27)+( 1
27- 1
32)
= 1
2 - 1
32
= 15
32
19.
解:原式= [(1+ 1
3 ) -( 1
3 + 1
4 ) +( 1
4 + 1
7 ) -( 1
7 - 1
8 ) -
( 1
8 + 1
11)+( 1
11+ 1
19)]×19
=(1+ 1
3 - 1
3 - 1
4 + 1
4 + 1
7 - 1
7 + 1
8 - 1
8 - 1
11 + 1
11 +
1
19)×19
= 20
19×19
= 20
20.
解:原式=[( 5
6 - 7
12+ 9
20-11
30+13
42-15
56)×7-1 3
8 ]×8
=[( 1
2 + 1
3 - 1
3 - 1
4 +…- 1
7 - 1
8 )×7-11
8 ]×8
=( 1
2 - 1
8 )×7×8-11
8 ×8
= 21-11
= 10
21.
解:原式= 1- 1
2 +1- 1
6 +1- 1
12+…+1- 1
110
= 1+1+1+…+1-( 1
2 + 1
6 + 1
12+ 1
20+…+ 1
110)
= 1×10-( 1
1×2+ 1
2×3+ 1
3×4+…+
1
10×11)
= 10-(1- 1
2 + 1
2 - 1
3 + 1
3 - 1
4 +…+ 1
10- 1
11)
= 10-(1- 1
11)
= 9 1
11
22.
解:原式= 4
3 +16
15+36
35+64
63+100
99
=(1+ 1
3 )+(1+ 1
15)+(1+ 1
35)+(1+ 1
63)+(1+ 1
99)
=(1+1+1+1)+( 1
3 + 1
15+ 1
35+ 1
65+ 1
99)
= 5+ 1
3 + 1
15+ 1
35+ 1
63+ 1
99
= 5+ 1
1×3+ 1
3×5+ 1
5×7+ 1
7×9+
1
9×11
= 5+ 1
2 ×(1- 1
3 + 1
3 - 1
5 + 1
5 - 1
7 + 1
7 - 1
9 + 1
9 - 1
11)
= 5+ 5
11
= 5 5
11
23.
解: 原式= (
30
1×3×5 +
30
3×5×7 +
30
5×7×9 + … +
30
29×31×33) -
(
1
1×3×5+
3
3×5×7+
5
5×7×9+…+
29
29×31×33)
= 15
2 ×(
4
1×3×5+
4
3×5×7+
4
5×7×9+…+
4
29×31×33) -
( 1
3×5+ 1
5×7+ 1
7×9+…+
1
31×33)
= 15
2 × ( 1
1×3 - 1
3×5 + 1
3×5 - 1
5×7 + 1
5×7 - 1
7×9 + … +
1
29×31-
1
31×33)- 1
2 ×( 1
3 - 1
5 + 1
5 - 1
7 + 1
7 - 1
9 +
…+ 1
31- 1
33)
= 15
2 ×( 1
3 -
1
31×33)- 1
2 ×( 1
3 - 1
33)
= 2395
1023
24.
解: 原式= (
1
1×2×3 -
1
2×3×4) + (
1
2×3×4 -
1
3×4×5) + … +
4
(
1
17×18×19-
1
18×19×20)
=
1
1×2×3-
1
18×19×20
= 1139
6840
25.
解:原式= 1
2
1×2+ 2
2
1×2+ 2
2
2×3+ 3
2
2×3+…+ 18
2
18×19+ 19
2
18×19+
19
2
19×20+ 20
2
19×20
= 1
2 +2+ 2
3 + 3
2 +…+18
19+19
18+19
20+20
19
= 38+19
20
= 38 19
20
26.
解:原式= 1+2×( 1
2 - 1
3 + 1
3 - 1
4 +…+ 1
100- 1
101)
= 1+2×( 1
2 - 1
101)
= 1+1- 2
101
= 1 99
101
27.
解:根据公式n(n+1)= 1
3 [n(n+1)(n+2) -(n-1)n(n+
1)]
原式= 1
3 ×(20×21×22-19×20×21+21×22×23-20×21×22
+22×23×24-21×22×23+…+99×100×101-98×99×
100)
= 1
3 ×(99×100×101-19×20×21)
= 330640
28.
解:n(n+1)(n+2)= 1
4 n(n+1)(n+2)(n+3)- 1
4 (n-1)n(n
+1)(n+2)
原式= 1
4 ×1×2×3×4- 1
4 ×0×1×2×3+ 1
4 ×2×3×4×5- 1
4 ×1
×2×3×4+ 1
4 ×3×4×5×6- 1
4 ×2×3×4×5+…+ 1
4 ×9×
10×11×12- 1
4 ×8×9×10×11
= 1
4 ×9×10×11×12
= 3×9×10×11
= 2970
29.
解:原式= 24 × ( 1
2×3 + 1
4×5 + … +
1
20×21) - 6 × (
1
1×2×3 +
1
2×3×5+…+
1
10×11×21)
= 24×( 1
2×3+ 1
4×5+…+
1
20×21)-24×(
1
2×4×3+
1
4×6×5+
…+
1
20×22×21)
= 24 × [( 1
2×3 -
1
2×4×3) + ( 1
4×5 -
1
4×6×5) + … +
(
1
20×21-
1
20×22×21)]
= 24×( 1
2×4+ 1
4×6+…+
1
20×22)
= 24× 1
2 ×( 1
2 - 1
4 + 1
4 - 1
6 +…+ 1
20 - 1
22)
= 12×( 1
2 - 1
22)
= 60
11
30.
解:原式= 100-1
1×2×3+100-2
2×3×4+100-3
3×4×5+…+
100-99
99×100×101
=
100
1×2×3- 1
2×3 +
100
2×3×4 - 1
3×4 +
100
3×4×5 - 1
4×5 + … +
100
99×100×101-
1
100×101
=
100
1×2×3+ 100
2×3×4+ 100
3×4×5+…+
100
99×100×101-( 1
2×3
+ 1
3×4+ 1
4×5+…+
1
100×101)
= 100× 1
2 ×( 1
1×2- 1
2×3+ 1
2×3- 1
3×4+ 1
3×4- 1
4×5+…+
1
99×100-
1
100×101)-( 1
2 - 1
3 + 1
3 - 1
4 + 1
4 - 1
5 +…
+ 1
100- 1
101)
= 100× 1
2 ×( 1
2 -
1
10100)-( 1
2 - 1
101)
= 25- 1
202- 1
2 + 1
101
= 24 51
101
专题10
公式法、换元法、
错位相减法、借一还一法
类型一 巧用公式法
1.
解:原式=(18+17)×(18-17) +(16+15) ×(16-15) +(14+
13)×(14-13)+…+(2+1)×(2-1)
= 18+17+16+15+14+13+…+2+1
=(18+1)×18
2
= 171
2.
解:原式=
1
(2+1)×(2-1)+
1
(4+1)×(4-1)+
1
(6+1)×(6-1)+
…+
1
(100+1)×(100-1)
= 1
1×3+ 1
3×5+ 1
5×7+…+
1
99×101
= 1
2 ×(1- 1
3 + 1
3 - 1
5 + 1
5 - 1
7 +…+ 1
99- 1
101)
= 1
2 ×(1- 1
101)
= 50
101
3.
解:原式=[1
2-( 1
2 )
2]×[1
2-( 1
3 )
2]×…×[1
2-( 1
10)
2]
=(1- 1
2 )×(1+ 1
2 ) ×(1- 1
3 ) ×(1+ 1
3 ) ×…×(1-
1
10)×(1+ 1
10)
= 1
2 × 3
2 × 2
3 × 4
3 ×…× 9
10×11
10
= 1
2 ×11
10
= 11
20
4.
解:原式= (2
2-1
2)+(4
2-3
2)+…+(100
2-99
2)
1+2+3+…+8+9+10+9+8+…+3+2+1
=(2+1)×(2-1)+(4+3)×(4-3)+…+(100+99)×(100-99)
1+2+3+…+8+9+10+9+8+…+3+2+1
=
1+2+3+4+…+99+100
1+2+3+…+8+9+10+9+8+…+3+2+1
=
100×(1+100)
2
9×(9+1)+10
= 101
2
5.
解:原式= 1999
2-1998
2+1997
2-1996
2+…+3
2-2
2+1
= (1999+ 1998) ×(1999- 1998) + (1997 + 1996) ×
(1997-1996)+…+(3+2)×(3-2)+1
= 1999+1998+1997+1996+…+3+2+1
= 1999×(1+1999)÷2
= 1999000
6.
解:原式= 1. 2345
2+2×1. 2345×0. 7655+0. 7655
2
=(1. 2345+0. 7655)
2
= 4
7.
解:原式=
(2022
2-2×2022+1)+2022
(2022
2-2×2022×2021+2021
2)+2022×2021
=
(2022-1)
2+2022
(2022-2021)
2+2022×2021
= 2021
2+2022
1+2022×2021
=
2021
2+2022
1+2021×(2021+1)
= 2021
2+2022
2021
2+2022
= 1
8.
解:因为
n
2
n
2-100n+5000
+
(100-n)
2
(100-n)
2-100(100-n)+5000
=
n
2
n
2-100n+5000
+10000-200n+n
2
n
2-100n+5000
= 2(n
2-100n+5000)
n
2-100n+5000
= 2
原式
=
(
1
2
1
2-100+5000
+
99
2
99
2-9900+5000
)
+ … +
(
49
2
49
2-4900+5000
+
51
2
51
2-5100+5000
)+
50
2
50
2-5000+5000
= 2+…+2+1
= 2×49+1
= 99
9.
解:原式=
(1+2013)×1007÷2
(2015+4027)×1007÷2
= 2014
6042
= 1
3
10.
解:原式= 1
500×[1+2×1+3+2×2+5+2×3+…+999+2×500]
= 1
500 ×[(1+ 3+ 5+…+ 999) + 2×(1+ 2+ 3+…+
500)]
= 1
500×[500×(1+999)
2
+2×500×(1+500)
2
]
= 1
500×(500×500+500×501)
= 500+501
= 1001
11.
解:原式= 1+(1+3)×2÷2
4
+(1+5)×3÷2
6
+…+(1+97)×49÷2
98
= 1+ 2
2 + 3
2 +…+49
2
= 1+(2+49)×48÷2
2
= 1+612
= 613
12.
解:原式= (2007- 2005) ×(2007
2 + 2007× 2005+ 2005
2 ) -
6×2006
2
5
= 2×[(2006+1)
2+(2006+1)×(2006-1)+(2006-
1)
2]-6×2006
2
= 2×[(2006
2+1
2 +2×2006×1+2006
2 -1
2 +2006
2 +
1
2-2×2006×1)-6×2006
2
= 2×(2006
2+1+2006
2-1+2006
2+1)-6×2006
2
= 2×(3×2006
2+1)-6×2006
2
= 6×2006
2+2-6×2006
2
= 2
解题技巧
立方差公式:a
3-b
3 =(a-b)(a
2+ab+b
2)。
13.
解:原式=(2+1)(2
2-2+1)
(2-1)(2
2+2+1)
×(3+1)(3
2-3+1)
(3-1)(3
2+3+1)
×
(4+1)(4
2-4+1)
(4-1)(4
2+4+1)
×…×(2016+1)
(2016-1)
(2016
2-2016+1)
(2016
2+2016+1)
×
(2017+1)
(2017-1)
(2017
2-2017+1)
(2017
2+2017+1)
=(2+1)(3
2-3+1)
(2-1)(2
2+2+1)
×(3+1)(4
2-4+1)
(3-1)(3
2+3+1)
×
(4+1)(5
2-5+1)
(4-1)(4
2+4+1)
×…×(2016+1)(2017
2-2017+1)
(2016-1)(2016
2+2016+1)
×
(2017+1)(2018
2-2018+1)
(2017-1)(2017
2+2017+1)
×
2
2-2+1
2018
2-2018+1
=(2016+1)×(2017+1)
(2-1)×(3-1)
×
2
2-2+1
2018
2-2018+1
= 2035153×3
4070307
= 2035153
1356769
解题技巧
先利用立方和公式:a
3 +b
3 = (a+b) (a
2 -ab+
b
2),再利用立方差公式:a
3-b
3 =(a-b)(a
2+ab+b
2)。
14.
解:原式=(1
2+2
2+3
2+…+21
2)-(1
2+2
2+3
2+…+14
2)
= 21×(21+1)×(2×21+1)
6
-14×(14+1)×(2×14+1)
6
= 3311-1015
= 2296
15.
解:原式= 4
2+(2×4)
2+(3×4)
2+…+(9×4)
2+(10×4)
2
= 4
2×(1
2+2
2+3
2+…+9
2+10
2)
= 16× 1
6 ×10×(10+1)×(2×10+1)
= 6160
16.
解:原式= 1×(1+1)+2×(2+1)+…+20×(20+1)
=
1
2+1+2
2+2+…+20
2+20
=(1
2+2
2+3
2+…+20
2)+(1+2+3+…+20)
= 1
6 ×20×(20+1)×(2×20+1)+(1+20)×20÷2
= 2870+210
= 3080
解题技巧
连续自然数的平方和公式:1
2 +2
2 +3
2 +…+n
2
= 1
6 ×n×(n+1)×(2n+1)。
17.
解:原式=(1+2+3+…+2017)
2
1+2+3+…+2017
= 2017×(2017+1)
2
= 2035153
18.
解:原式=(1
3+2
3+…+100
3)-(2
3+4
3+…+100
3)
= 100
2×101
2
4
-2
3×(1
3+2
3+…+50
3)
= 100
2×101
2
4
-2
3×50
2×51
2
4
= 50
2×(101
2-2×51
2)
= 12497500
解题技巧
连续自然数立方和公式:1
3 +2
3 +3
3 +…+n
3 =
(1+2+3+…+n)
2 =n
2(n+1)
2
4
。
类型二 换元法
19.
解:令1+0. 12+0. 23=a,0. 12+0. 23=b,
则a-b= 1,
原式=a×(b+0. 34)-(a+0. 34)×b
=ab+0. 34a-ab-0. 34b
= 0. 34×(a-b)
= 0. 34
20.
解:令1
2 + 1
3 + 1
4 + 1
5 =a, 1
3 + 1
4 =b,
原式=(1+b)×a-(1+a)×b
=a+ab-b-ab
=a-b
=( 1
2 + 1
3 + 1
4 + 1
5 )-( 1
3 + 1
4 )
= 1
2 + 1
5
= 7
10
21.
解:设a= 1234567891
原式=
a-1
a
2-(a-1)(a+1)
=
a-1
a
2-(a
2-1)
=
a-1
a
2-a
2+1
=a-1
= 1234567890
22.
解:令1
2 + 1
3 +…+ 1
2023 =a, 1
2 + 1
3 +…+ 1
2022 =b,
则a-b=
1
2023,原式=(1-b)×a-(1-a)×b
=a-ab-b+ab
=a-b
=
1
2023
23.
解:设3+
1
4+
1
…+ 1
2009
=a,
原式=
1
2+ 1
a
+
1
1+
1
1+ 1
a
=
1
2a+1
a
+
1
1+ 1
a+1
a
=
a
2a+1+
1
2a+1
a+1
=
a
2a+1+ a+1
2a+1
= 1
类型三 错位相减法
24.
解:令Sn = 1-2+2
2-2
3+…+2
98-2
99+2
100①,2Sn = 2-2
2+2
3-
2
4+…+2
99-2
100+2
101②,
①+②得:Sn+2Sn = 1+2
101,即3Sn = 1+2
101,Sn = 1+2
101
3
,所以
1-2+2
2-2
3+…+2
98-2
99+2
100 = 1+2
101
3
。
25.
解:令S= 1+ 1
3 + 1
3
2 + 1
3
3 + 1
3
4 ,
则3S= 3+1+ 1
3 + 1
3
2 + 1
3
3 ,3S-S= 2S= 3-
1
3
4 ,
所以S= 242
81 ÷2= 121
81 ,原式= 121
81 。
类型四 借一还一法
26.
解:原式= (10- 1) +(100- 1) +(1000- 1) +(10000- 1) +
(100000-1)+(1000000-1)
= 10+100+1000+10000+100000+1000000-6
= 1111110-6
= 1111104
27.
解:原式=( 1
2 + 1
4 + 1
8 + 1
16+ 1
32+ 1
64+ 1
128+ 1
128)- 1
128
=( 1
2 + 1
2 )- 1
128
= 127
128
28.
解:原式= (9 1
5 + 4
5 ) + (99 1
5 + 4
5 ) + ( 999 1
5 + 4
5 ) +
(9999 1
5 + 4
5 )- 4
5 ×5
= 10+100+1000+10000-4
= 11110-4
= 11106
29.
解:原式= (2- 0. 1) +(20- 0. 01) × 2+(200- 0. 001) × 3+
(2000-0. 0001)×4+(20000-0. 00001)×5
= 2-0. 1+20×2-0. 01×2+200×3-0. 001×3+2000×
4-0. 0001×4+20000×5-0. 00001×5
= 2-0. 1+40-0. 02+600-0. 003+ 8000- 0. 0004+
100000-0. 00005
=(2+40+600+8000+100000)-(0. 1+0. 02+0. 003+
0. 0004+0. 00005)
= 108642-0. 12345
= 108641. 87655
6
