根号表
√1 = 1
√2 = 1.41421√3 = 1.73205√4 = 2
√5 = 2.23607√6 = 2.44949√7 = 2.64575√8 = 2.82843√9 = 3
√10 = 3.16228√11 = 3.31662√12 = 3.4641√13 = 3.60555√14 = 3.74166√15 = 3.87298√16 = 4
√17 = 4.12311√18 = 4.24264√19 = 4.3589√20 = 4.47214√21 = 4.58258 √22 = 4.69042√23 = 4.79583√24 = 4.89898√25 = 5
√26 = 5.09902√27 = 5.19615√28 = 5.2915√29 = 5.38516√30 = 5.47723√31 = 5.56776√32 = 5.65685√33 = 5.74456√34 = 5.83095√35 = 5.91608√36 = 6
√37 = 6.08276√38 = 6.16441√39 = 6.245
√40 = 6.32456√41 = 6.40312 √42 = 6.48074√43 = 6.55744√44 = 6.63325√45 = 6.7082√46 = 6.78233√47 = 6.85565√48 = 6.9282√49 = 7
√50 = 7.07107√51 = 7.14143
√52 = 7.2111
√53 = 7.28011
√54 = 7.34847
√55 = 7.4162
√56 = 7.48331
√57 = 7.54983
√58 = 7.61577
√59 = 7.68115
√60 = 7.74597
√61 = 7.81025
√62 = 7.87401
√63 = 7.93725
√64 = 8
√65 = 8.06226
√66 = 8.12404
√67 = 8.18535
√68 = 8.24621
√69 = 8.30662
√70 = 8.3666
√71 = 8.42615
√72 = 8.48528
√73 = 8.544
√74 = 8.60233
√75 = 8.66025
√76 = 8.7178
√77 = 8.77496
√78 = 8.83176
√79 = 8.88819
√80 = 8.94427
√81 = 9
√82 = 9.05539
√83 = 9.11043
√84 = 9.16515
√85 = 9.21954
√86 = 9.27362
√87 = 9.32738
√88 = 9.38083
√89 = 9.43398
√90 = 9.48683
√91 = 9.53939
√92 = 9.59166
√93 = 9.64365
√94 = 9.69536
√95 = 9.74679
√96 = 9.79796
√97 = 9.84886
√98 = 9.89949
√99 = 9.94987
√100 = 10
√101 = 10.0499
√102 = 10.0995
√103 = 10.1489
√104 = 10.198
√105 = 10.247
√106 = 10.2956
√107 = 10.3441
√108 = 10.3923
√109 = 10.4403
√110 = 10.4881
√111 = 10.5357
√112 = 10.583
√113 = 10.6301
√114 = 10.6771
√115 = 10.7238
√116 = 10.7703
√117 = 10.8167
√118 = 10.8628
√119 = 10.9087
√120 = 10.9545
√121 = 11
√122 = 11.0454
√123 = 11.0905
√124 = 11.1355
√125 = 11.1803
√126 = 11.225
√127 = 11.2694
√128 = 11.3137
√129 = 11.3578
√130 = 11.4018
√131 = 11.4455
√132 = 11.4891
√133 = 11.5326
√134 = 11.5758
√135 = 11.619
√136 = 11.6619
√137 = 11.7047
√138 = 11.7473
√139 = 11.7898
√140 = 11.8322
√141 = 11.8743
√142 = 11.9164
√143 = 11.9583
√144 = 12
√145 = 12.0416
√146 = 12.083
√147 = 12.1244
√148 = 12.1655
√149 = 12.2066
√150 = 12.2474
√151 = 12.2882
√152 = 12.3288
√153 = 12.3693
√154 = 12.4097
√155 = 12.4499
√156 = 12.49
√157 = 12.53
√158 = 12.5698
√159 = 12.6095
√160 = 12.6491
√161 = 12.6886
√162 = 12.7279
√163 = 12.7671
√164 = 12.8062
√165 = 12.8452
√166 = 12.8841
√167 = 12.9228
√168 = 12.9615
√169 = 13
√170 = 13.0384
√171 = 13.0767
√172 = 13.1149
√173 = 13.1529
√174 = 13.1909
√175 = 13.2288
√176 = 13.2665
√177 = 13.3041
√178 = 13.3417
√179 = 13.3791
√180 = 13.4164
√181 = 13.4536
√182 = 13.4907
√183 = 13.5277
√184 = 13.5647
√185 = 13.6015
√186 = 13.6382
√187 = 13.6748
√188 = 13.7113
√189 = 13.7477
√190 = 13.784
√191 = 13.8203
√192 = 13.8564
√193 = 13.8924
√194 = 13.9284
√195 = 13.9642
√196 = 14
√197 = 14.0357
√198 = 14.0712
√199 = 14.1067
√200 = 14.1421
√201 = 14.1774√202 = 14.2127 √203 = 14.2478√204 = 14.2829√205 = 14.3178√206 = 14.3527√207 = 14.3875√208 = 14.4222√209 = 14.4568√210 = 14.4914√211 = 14.5258√212 = 14.5602√213 = 14.5945√214 = 14.6287√215 = 14.6629√216 = 14.6969√217 = 14.7309√218 = 14.7648√219 = 14.7986 √220 = 14.8324√221 = 14.8661√222 = 14.8997√223 = 14.9332√224 = 14.9666√225 = 15
√226 = 15.0333√227 = 15.0665√228 = 15.0997√229 = 15.1327√230 = 15.1658√231 = 15.1987√232 = 15.2315√233 = 15.2643√234 = 15.2971√235 = 15.3297√236 = 15.3623√237 = 15.3948 √238 = 15.4272√239 = 15.4596√240 = 15.4919√241 = 15.5242√242 = 15.5563√243 = 15.5885√244 = 15.6205√245 = 15.6525√246 = 15.6844√247 = 15.7162√248 = 15.748√249 = 15.7797√250 = 15.8114√251 = 15.843
√252 = 15.8745
√253 = 15.906
√254 = 15.9374
√255 = 15.9687
√256 = 16
√257 = 16.0312
√258 = 16.0624
√259 = 16.0935
√260 = 16.1245
√261 = 16.1555
√262 = 16.1864
√263 = 16.2173
√264 = 16.2481
√265 = 16.2788
√266 = 16.3095
√267 = 16.3401
√268 = 16.3707
√269 = 16.4012
√270 = 16.4317
√271 = 16.4621
√272 = 16.4924
√273 = 16.5227
√274 = 16.5529
√275 = 16.5831
√276 = 16.6132
√277 = 16.6433
√278 = 16.6733
√279 = 16.7033
√280 = 16.7332
√281 = 16.7631
√282 = 16.7929
√283 = 16.8226
√284 = 16.8523
√285 = 16.8819
√286 = 16.9115
√287 = 16.9411
√288 = 16.9706
√289 = 17
√290 = 17.0294
√291 = 17.0587
√292 = 17.088
√293 = 17.1172
√294 = 17.1464
√295 = 17.1756
√296 = 17.2047
√297 = 17.2337
√298 = 17.2627
√299 = 17.2916
√300 = 17.3205
√301 = 17.3494
√302 = 17.3781
√303 = 17.4069
√304 = 17.4356
√305 = 17.4642
√306 = 17.4929
√307 = 17.5214
√308 = 17.5499
√309 = 17.5784
√310 = 17.6068
√311 = 17.6352
√312 = 17.6635
√313 = 17.6918
√314 = 17.72
√315 = 17.7482
√316 = 17.7764
√317 = 17.8045
√318 = 17.8326
√319 = 17.8606
√320 = 17.8885
√321 = 17.9165
√322 = 17.9444
√323 = 17.9722
√324 = 18
√325 = 18.0278
√326 = 18.0555
√327 = 18.0831
√328 = 18.1108
√329 = 18.1384
√330 = 18.1659
√331 = 18.1934
√332 = 18.2209
√333 = 18.2483
√334 = 18.2757
√335 = 18.303
√336 = 18.3303
√337 = 18.3576
√338 = 18.3848
√339 = 18.412
√340 = 18.4391
√341 = 18.4662
√342 = 18.4932
√343 = 18.5203
√344 = 18.5472
√345 = 18.5742
√346 = 18.6011
√347 = 18.6279
√348 = 18.6548
√349 = 18.6815
√350 = 18.7083
√351 = 18.735
√352 = 18.7617
√353 = 18.7883
√354 = 18.8149
√355 = 18.8414
√356 = 18.868
√357 = 18.8944
√358 = 18.9209
√359 = 18.9473
√360 = 18.9737
√361 = 19
√362 = 19.0263
√363 = 19.0526
√364 = 19.0788
√365 = 19.105
√366 = 19.1311
√367 = 19.1572
√368 = 19.1833
√369 = 19.2094
√370 = 19.2354
√371 = 19.2614
√372 = 19.2873
√373 = 19.3132
√374 = 19.3391
√375 = 19.3649
√376 = 19.3907
√377 = 19.4165
√378 = 19.4422
√379 = 19.4679
√380 = 19.4936
√381 = 19.5192
√382 = 19.5448
√383 = 19.5704
√384 = 19.5959
√385 = 19.6214
√386 = 19.6469
√387 = 19.6723
√388 = 19.6977
√389 = 19.7231
√390 = 19.7484
√391 = 19.7737
√392 = 19.799
√393 = 19.8242
√394 = 19.8494
√395 = 19.8746
√396 = 19.8997
√397 = 19.9249
√398 = 19.9499
√399 = 19.975
√400 = 20
√401 = 20.025√402 = 20.0499√403 = 20.0749√404 = 20.0998√405 = 20.1246√406 = 20.1494√407 = 20.1742√408 = 20.199√409 = 20.2237√410 = 20.2485 √411 = 20.2731√412 = 20.2978√413 = 20.3224√414 = 20.347√415 = 20.3715√416 = 20.3961√417 = 20.4206√418 = 20.445√419 = 20.4695√420 = 20.4939√421 = 20.5183√422 = 20.5426√423 = 20.567√424 = 20.5913√425 = 20.6155√426 = 20.6398√427 = 20.664√428 = 20.6882√429 = 20.7123√430 = 20.7364√431 = 20.7605√432 = 20.7846√433 = 20.8087√434 = 20.8327√435 = 20.8567√436 = 20.8806√437 = 20.9045√438 = 20.9284√439 = 20.9523√440 = 20.9762√441 = 21
√442 = 21.0238√443 = 21.0476√444 = 21.0713√445 = 21.095 √446 = 21.1187√447 = 21.1424√448 = 21.166√449 = 21.1896√450 = 21.2132√451 = 21.2368
√452 = 21.2603
√453 = 21.2838
√454 = 21.3073
√455 = 21.3307
√456 = 21.3542
√457 = 21.3776
√458 = 21.4009
√459 = 21.4243
√460 = 21.4476
√461 = 21.4709
√462 = 21.4942
√463 = 21.5174
√464 = 21.5407
√465 = 21.5639
√466 = 21.587
√467 = 21.6102
√468 = 21.6333
√469 = 21.6564
√470 = 21.6795
√471 = 21.7025
√472 = 21.7256
√473 = 21.7486
√474 = 21.7715
√475 = 21.7945
√476 = 21.8174
√477 = 21.8403
√478 = 21.8632
√479 = 21.8861
√480 = 21.9089
√481 = 21.9317
√482 = 21.9545
√483 = 21.9773
√484 = 22
√485 = 22.0227
√486 = 22.0454
√487 = 22.0681
√488 = 22.0907
√489 = 22.1133
√490 = 22.1359
√491 = 22.1585
√492 = 22.1811
√493 = 22.2036
√494 = 22.2261
√495 = 22.2486
√496 = 22.2711
√497 = 22.2935
√498 = 22.3159
√499 = 22.3383
√500 = 22.3607
√501 = 22.383
√502 = 22.4054
√503 = 22.4277
√504 = 22.4499
√505 = 22.4722
√506 = 22.4944
√507 = 22.5167
√508 = 22.5389
√509 = 22.561
√510 = 22.5832
√511 = 22.6053
√512 = 22.6274
√513 = 22.6495
√514 = 22.6716
√515 = 22.6936
√516 = 22.7156
√517 = 22.7376
√518 = 22.7596
√519 = 22.7816
√520 = 22.8035
√521 = 22.8254
√522 = 22.8473
√523 = 22.8692
√524 = 22.891
√525 = 22.9129
√526 = 22.9347
√527 = 22.9565
√528 = 22.9783
√529 = 23
√530 = 23.0217
√531 = 23.0434
√532 = 23.0651
√533 = 23.0868
√534 = 23.1084
√535 = 23.1301
√536 = 23.1517
√537 = 23.1733
√538 = 23.1948
√539 = 23.2164
√540 = 23.2379
√541 = 23.2594
√542 = 23.2809
√543 = 23.3024
√544 = 23.3238
√545 = 23.3452
√546 = 23.3666
√547 = 23.388
√548 = 23.4094
√549 = 23.4307
√550 = 23.4521
√551 = 23.4734
√552 = 23.4947
√553 = 23.516
√554 = 23.5372
√555 = 23.5584
√556 = 23.5797
√557 = 23.6008
√558 = 23.622
√559 = 23.6432
√560 = 23.6643
√561 = 23.6854
√562 = 23.7065
√563 = 23.7276
√564 = 23.7487
√565 = 23.7697
√566 = 23.7908
√567 = 23.8118
√568 = 23.8328
√569 = 23.8537
√570 = 23.8747
√571 = 23.8956
√572 = 23.9165
√573 = 23.9374
√574 = 23.9583
√575 = 23.9792
√576 = 24
√577 = 24.0208
√578 = 24.0416
√579 = 24.0624
√580 = 24.0832
√581 = 24.1039
√582 = 24.1247
√583 = 24.1454
√584 = 24.1661
√585 = 24.1868
√586 = 24.2074
√587 = 24.2281
√588 = 24.2487
√589 = 24.2693
√590 = 24.2899
√591 = 24.3105
√592 = 24.3311
√593 = 24.3516
√594 = 24.3721
√595 = 24.3926
√596 = 24.4131
√597 = 24.4336
√598 = 24.454
√599 = 24.4745
√600 = 24.4949
√601 = 24.5153√602 = 24.5357√603 = 24.5561√604 = 24.5764√605 = 24.5967√606 = 24.6171√607 = 24.6374√608 = 24.6577√609 = 24.6779√610 = 24.6982√611 = 24.7184√612 = 24.7386√613 = 24.7588√614 = 24.779√615 = 24.7992√616 = 24.8193√617 = 24.8395 √618 = 24.8596√619 = 24.8797√620 = 24.8998√621 = 24.9199√622 = 24.9399√623 = 24.96
√624 = 24.98
√625 = 25
√626 = 25.02
√627 = 25.04
√628 = 25.0599√629 = 25.0799√630 = 25.0998√631 = 25.1197√632 = 25.1396√633 = 25.1595√634 = 25.1794√635 = 25.1992 √636 = 25.219√637 = 25.2389√638 = 25.2587√639 = 25.2784√640 = 25.2982√641 = 25.318√642 = 25.3377√643 = 25.3574√644 = 25.3772√645 = 25.3969√646 = 25.4165√647 = 25.4362√648 = 25.4558√649 = 25.4755√650 = 25.4951√651 = 25.5147
√652 = 25.5343
√653 = 25.5539
√654 = 25.5734
√655 = 25.593
√656 = 25.6125
√657 = 25.632
√658 = 25.6515
√659 = 25.671
√660 = 25.6905
√661 = 25.7099
√662 = 25.7294
√663 = 25.7488
√664 = 25.7682
√665 = 25.7876
√666 = 25.807
√667 = 25.8263
√668 = 25.8457
√669 = 25.865
√670 = 25.8844
√671 = 25.9037
√672 = 25.923
√673 = 25.9422
√674 = 25.9615
√675 = 25.9808
√676 = 26
√677 = 26.0192
√678 = 26.0384
√679 = 26.0576
√680 = 26.0768
√681 = 26.096
√682 = 26.1151
√683 = 26.1343
√684 = 26.1534
√685 = 26.1725
√686 = 26.1916
√687 = 26.2107
√688 = 26.2298
√689 = 26.2488
√690 = 26.2679
√691 = 26.2869
√692 = 26.3059
√693 = 26.3249
√694 = 26.3439
√695 = 26.3629
√696 = 26.3818
√697 = 26.4008
√698 = 26.4197
√699 = 26.4386
√700 = 26.4575
√701 = 26.4764
√702 = 26.4953
√703 = 26.5141
√704 = 26.533
√705 = 26.5518
√706 = 26.5707
√707 = 26.5895
√708 = 26.6083
√709 = 26.6271
√710 = 26.6458
√711 = 26.6646
√712 = 26.6833
√713 = 26.7021
√714 = 26.7208
√715 = 26.7395
√716 = 26.7582
√717 = 26.7769
√718 = 26.7955
√719 = 26.8142
√720 = 26.8328
√721 = 26.8514
√722 = 26.8701
√723 = 26.8887
√724 = 26.9072
√725 = 26.9258
√726 = 26.9444
√727 = 26.9629
√728 = 26.9815
√729 = 27
√730 = 27.0185
√731 = 27.037
√732 = 27.0555
√733 = 27.074
√734 = 27.0924
√735 = 27.1109
√736 = 27.1293
√737 = 27.1477
√738 = 27.1662
√739 = 27.1846
√740 = 27.2029
√741 = 27.2213
√742 = 27.2397
√743 = 27.258
√744 = 27.2764
√745 = 27.2947
√746 = 27.313
√747 = 27.3313
√748 = 27.3496
√749 = 27.3679
√750 = 27.3861
√751 = 27.4044
√752 = 27.4226
√753 = 27.4408
√754 = 27.4591
√755 = 27.4773
√756 = 27.4955
√757 = 27.5136
√758 = 27.5318
√759 = 27.55
√760 = 27.5681
√761 = 27.5862
√762 = 27.6043
√763 = 27.6225
√764 = 27.6405
√765 = 27.6586
√766 = 27.6767
√767 = 27.6948
√768 = 27.7128
√769 = 27.7308
√770 = 27.7489
√771 = 27.7669
√772 = 27.7849
√773 = 27.8029
√774 = 27.8209
√775 = 27.8388
√776 = 27.8568
√777 = 27.8747
√778 = 27.8927
√779 = 27.9106
√780 = 27.9285
√781 = 27.9464
√782 = 27.9643
√783 = 27.9821
√784 = 28
√785 = 28.0179
√786 = 28.0357
√787 = 28.0535
√788 = 28.0713
√789 = 28.0891
√790 = 28.1069
√791 = 28.1247
√792 = 28.1425
√793 = 28.1603
√794 = 28.178
√795 = 28.1957
√796 = 28.2135
√797 = 28.2312
√798 = 28.2489
√799 = 28.2666
√800 = 28.2843
√801 = 28.3019√802 = 28.3196√803 = 28.3373√804 = 28.3549√805 = 28.3725√806 = 28.3901√807 = 28.4077√808 = 28.4253 √809 = 28.4429√810 = 28.4605√811 = 28.4781√812 = 28.4956√813 = 28.5132√814 = 28.5307√815 = 28.5482√816 = 28.5657√817 = 28.5832√818 = 28.6007√819 = 28.6182√820 = 28.6356√821 = 28.6531√822 = 28.6705√823 = 28.688√824 = 28.7054√825 = 28.7228 √826 = 28.7402√827 = 28.7576√828 = 28.775√829 = 28.7924√830 = 28.8097√831 = 28.8271√832 = 28.8444√833 = 28.8617√834 = 28.8791√835 = 28.8964√836 = 28.9137√837 = 28.931√838 = 28.9482√839 = 28.9655√840 = 28.9828√841 = 29
√842 = 29.0172 √843 = 29.0345√844 = 29.0517√845 = 29.0689√846 = 29.0861√847 = 29.1033√848 = 29.1204√849 = 29.1376√850 = 29.1548√851 = 29.1719
√852 = 29.189
√853 = 29.2062
√854 = 29.2233
√855 = 29.2404
√856 = 29.2575
√857 = 29.2746
√858 = 29.2916
√859 = 29.3087
√860 = 29.3258
√861 = 29.3428
√862 = 29.3598
√863 = 29.3769
√864 = 29.3939
√865 = 29.4109
√866 = 29.4279
√867 = 29.4449
√868 = 29.4618
√869 = 29.4788
√870 = 29.4958
√871 = 29.5127
√872 = 29.5296
√873 = 29.5466
√874 = 29.5635
√875 = 29.5804
√876 = 29.5973
√877 = 29.6142
√878 = 29.6311
√879 = 29.6479
√880 = 29.6648
√881 = 29.6816
√882 = 29.6985
√883 = 29.7153
√884 = 29.7321
√885 = 29.7489
√886 = 29.7658
√887 = 29.7825
√888 = 29.7993
√889 = 29.8161
√890 = 29.8329
√891 = 29.8496
√892 = 29.8664
√893 = 29.8831
√894 = 29.8998
√895 = 29.9166
√896 = 29.9333
√897 = 29.95
√898 = 29.9666
√899 = 29.9833
√900 = 30
√901 = 30.0167
√902 = 30.0333
√903 = 30.05
√904 = 30.0666
√905 = 30.0832
√906 = 30.0998
√907 = 30.1164
√908 = 30.133
√909 = 30.1496
√910 = 30.1662
√911 = 30.1828
√912 = 30.1993
√913 = 30.2159
√914 = 30.2324
√915 = 30.249
√916 = 30.2655
√917 = 30.282
√918 = 30.2985
√919 = 30.315
√920 = 30.3315
√921 = 30.348
√922 = 30.3645
√923 = 30.3809
√924 = 30.3974
√925 = 30.4138
√926 = 30.4302
√927 = 30.4467
√928 = 30.4631
√929 = 30.4795
√930 = 30.4959
√931 = 30.5123
√932 = 30.5287
√933 = 30.545
√934 = 30.5614
√935 = 30.5778
√936 = 30.5941
√937 = 30.6105
√938 = 30.6268
√939 = 30.6431
√940 = 30.6594
√941 = 30.6757
√942 = 30.692
√943 = 30.7083
√944 = 30.7246
√945 = 30.7409
√946 = 30.7571
√947 = 30.7734
√948 = 30.7896
√949 = 30.8058
√950 = 30.8221
√951 = 30.8383
√952 = 30.8545
√953 = 30.8707
√954 = 30.8869
√955 = 30.9031
√956 = 30.9192
√957 = 30.9354
√958 = 30.9516
√959 = 30.9677
√960 = 30.9839
√961 = 31
√962 = 31.0161
√963 = 31.0322
√964 = 31.0483
√965 = 31.0644
√966 = 31.0805
√967 = 31.0966
√968 = 31.1127
√969 = 31.1288
√970 = 31.1448
√971 = 31.1609
√972 = 31.1769
√973 = 31.1929
√974 = 31.209
√975 = 31.225
√976 = 31.241
√977 = 31.257
√978 = 31.273
√979 = 31.289
√980 = 31.305
√981 = 31.3209
√982 = 31.3369
√983 = 31.3528
√984 = 31.3688
√985 = 31.3847
√986 = 31.4006
√987 = 31.4166
√988 = 31.4325
√989 = 31.4484
√990 = 31.4643
√991 = 31.4802
√992 = 31.496
√993 = 31.5119
√994 = 31.5278
√995 = 31.5436
√996 = 31.5595
√997 = 31.5753
√998 = 31.5911
√999 = 31.607
√1000 = 31.6228
自然数平方数列和立方数列求和公式
自然数平方数列和立方数列求和公式怎么推导？即： (1) 1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)/6 (2) 1^3+2^3+3^3+……+n^3=[n(n+1)/2]^2 推导过程如下： 一. 1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)/6 利用立方差公式 n^3-(n-1)^3=1*[n^2+(n-1)^2+n(n-1)] =n^2+(n-1)^2+n^2-n =2*n^2+(n-1)^2-n 2^3-1^3=2*2^2+1^2-2 3^3-2^3=2*3^2+2^2-3 4^3-3^3=2*4^2+3^2-4 ...... n^3-(n-1)^3=2*n^2+(n-1)^2-n 各等式全相加 n^3-1^3=2*(2^2+3^2+...+n^2)+[1^2+2^2+...+(n-1)^2]-(2+3+4+...+n) n^3-1=2*(1^2+2^2+3^2+...+n^2)-2+[1^2+2^2+...+(n-1)^2+n^2]-n^2-(2+3+4+... +n) n^3-1=3*(1^2+2^2+3^2+...+n^2)-2-n^2-(1+2+3+...+n)+1 n^3-1=3(1^2+2^2+...+n^2)-1-n^2-n(n+1)/2 3(1^2+2^2+...+n^2)=n^3+n^2+n(n+1)/2=(n/2)(2n^2+2n+n+1) =(n/2)(n+1)(2n+1) 故：1^2+2^2+3^2+...+n^2=n(n+1)(2n+1)/6 二. 1^3+2^3+3^3+……+n^3=[n(n+1)/2]^2 证明如下： (n+1)^4-n^4=[(n+1)^2+n^2][(n+1)^2-n^2] =(2n^2+2n+1)(2n+1)
平方、立方表
1*1=1 2*2=4 3*3=9 4*4=16 5*5=25 6*6=36 7*7=49 8*8=64 9*9=81 10*10=100 11*11=121 12*12=144 13*13=169 14*14=196 15*15=225 16*16=256 17*17=289 18*18=324 19*19=361 20*20=400 21*21=441 22*22=484 23*23=529 24*24=576 25*25=625 26*26=676 27*27=729 28*28=784 29*29=841 30*30=900 31*31=961 32*32=1024 33*33=1089 34*34=1156 35*35=1225 36*36=1296 37*37=1369 38*38=1444 39*39=1521 40*40=1600 41*41=1681 42*42=1764 43*43=1849 44*44=1936 45*45=2025 46*46=2116 47*47=2209 48*48=2304 49*49=2401 50*50=2500 51*51=2601 52*52=2704 53*53=2809 54*54=2916 55*55=3025 56*56=3136 57*57=3249 58*58=3364 59*59=3481 60*60=3600 61*61=3721 62*62=3844 63*63=3969 64*64=4096 65*65=4225 66*66=4356 1——100的平方表
67*67=4489 68*68=4624 69*69=4761 70*70=4900 71*71=5041 72*72=5184 73*73=5329 74*74=5476 75*75=5625 76*76=5776 77*77=5929 78*78=6084 79*79=6241 80*80=6400 81*81=6561 82*82=6724 83*83=6889 84*84=7056 85*85=7225 86*86=7396 87*87=7569 88*88=7744 89*89=792190*90=8100 91*91=8281 92*92=8464 93*93=8649 94*94=8836 95*95=9025 96*96=9216 97*97=9409 98*98=9604 99*99=9801 100*100=10000
前n个自然数的平方和及证明
帕斯卡与前n 个自然数的平方和 十七世纪的法国数学家帕斯卡（Pascal B.，1623.6.19~1662.8.19）想出了一个新的很妙的方法能求出前n 个自然数的平方和。这个方法是这样的： 利用和的立方公式，我们有 （n ＋1）3＝n 3＋3n 2＋3n ＋1， 移项可得 （n ＋1）3 －n 3＝3n 2＋3n ＋1， 此式对于任何自然数n 都成立。 依次把n ＝1,2，3，…，n －1，n 代入上式可得 23 －13＝3?12＋3?1＋1， 33 －23＝3?22＋3?2＋1， 43 －33＝3?32＋3?3＋1， …………………………… n 3－（n －1）3＝3（n －1）2＋3（n －1）＋1， （n ＋1）3 －n 3＝3n 2＋3n ＋1， 把这n 个等式的左边与右边对应相加，则n 个等式的左边各项两两相消，最后只剩下（n ＋1）3 － 1；而n 个等式的右边各项，我们把它们按三列相加，提取公因数后，第一列出现我们所要计算的前n 个自然数的平方和，第二列出现我们在上一段已经算过的前n 个自然数的和，第三列是n 个1。因而我们得到 （n ＋1）3 －1＝3S n ＋ 2)1(3+n n ＋n ， 现在这里S n ＝12＋22＋…＋n 2。 对这个结果进行恒等变形可得 n 3＋3n 2＋3n ＝3S n ＋ 2)1(3+n n ＋n ， 2n 3＋6n 2＋6n ＝6S n ＋3n 2＋3n ＋2n 移项、合并同类项可得 6S n ＝2n 3＋3n 2＋n ＝n （n ＋1）（2n ＋1）， ∴S n ＝ 61n （n ＋1）（2n ＋1）， 即 12＋22＋32＋…＋n 2＝6 1n （n ＋1）（2n ＋1）。 这个方法把所要计算的前n 个自然数的平方和与已知的前n 个自然数的和及其它一些已知量通过一个方程联系起来，然后解方程求出所希望得到的公式，确实是很妙的。
自然数的和,平方和,立方和
For personal use only in study and research; not for commercial use 求：①自然数（一次方）的和，即：n ++++ 321 ②自然数平方（二次方）的和，即：2222321n ++++ ③自然数立方（三次方）的和，即：3333321n ++++ 求①式可用2)1(+n 来计算；求②式可用3)1(+n 来计算；求③式可用4)1(+n 来计算 ① ∵12)1(22++=+n n n ∴ 1121222+?+= …… 将以上等式两边相加得： ∴ n ++++ 3212 )1(+= n n ② ∵3)1(+n = 13323+++n n n ∴ 1131312233+?+?+= …… 3)1(+n = 13323+++n n n 将以上等式两边相加得： )321(32222n ++++? = 3)1(+n —?? ????++?+n n n 2)1(313 ∴ 2222321n ++++ = 6 )12)(1(++n n n ③ 用同样的方法，可得： 3333321n ++++ = 4)1(22+n n = 22)1(?? ? ??+n n 自然数的立方和等于自然数和的平方。 利用上面三个结论，我们就可以计算下面数列的和了。 ④ )321()321()21(1n +++++++++++ ∵n ++++ 3212)1(+=n n = n n 2 1212+
∴ 12 112112?+?= …… n ++++ 321 = n n 2 1212+ 将上面各式左右两边分别相加，得： )321()321()21(1n +++++++++++ = )321(2 12222n ++++ = ?? ? ??++++2)1(6)12)(1(21n n n n n = 6 )2)(1(++n n n ⑤ )1(433221+++?+?+?n n = 3 )2)(1(++n n n ⑥ )2)(1(543432321++++??+??+??n n n = 4)3)(2)(1(+++n n n n
100以内的平方数与立方数
平方表 平方根平方数平方根平方数平方根平方数平方根平方数 1 1 26 676 51 2601 76 5776 2 4 27 729 52 2704 77 5929 3 9 28 78 4 53 2809 78 6084 4 16 29 841 54 2916 79 6241 5 25 30 900 55 3025 80 6400 6 36 31 961 56 3136 81 6561 7 49 32 1024 57 3249 82 6724 8 64 33 1089 58 3364 83 6889 9 81 34 1156 59 3481 84 7056 10 100 35 1225 60 3600 85 7225 11 121 36 1296 61 3721 86 7396 12 144 37 1369 62 3844 87 7569 13 169 38 1444 63 3969 88 7744 14 196 39 1521 64 4096 89 7921 15 225 40 1600 65 4225 90 8100 16 256 41 1681 66 4356 91 8281 17 289 42 1764 67 4489 92 8464 18 324 43 1849 68 4624 93 8649 19 361 44 1936 69 4761 94 8836 20 400 45 2025 70 4900 95 9025 21 441 46 2116 71 5041 96 9216 22 484 47 2209 72 5184 97 9409 23 529 48 2304 73 5329 98 9604 24 576 49 2401 74 5476 99 9801 25 625 50 2500 75 5625 100 10000
常用平方、立方表
=1 =4 =9 =16 =25 =36 =49 =64 =81 =100 =121 =144 =169 =196 =225 =256 =289 =324 =361 =400 2=441 =484 =529 =576 =625 =676 =729 =784 =841 =900 =961 =1024 =1089 =1156 =1225 =1296 =1369 =1444 =1521 =1600 =1681 =1764 =1849 =1936 =2025 =2116 =2209 =2304 =2401 =2500 =2601 =2704 =2809 =2916 =3025 =3136 =3249 =3364 =3481 =3600 =3721 =3844 =3969 =4096 =4225 =4356 =4489 =4624 =4761 =4900 =5041 =5184 =5329 =5476 =5625 =5776 =5929 =6084 =6241 =6400 =6561 =6724 1——100的平方表
=6889 =7056 =7225 =7396 =7569 =7744 =7921 =8100 =8281 =8464 =8649 =8836 =9025 =9216 =9409 =9604 =9801 =10000 =1 =8 =27=64 =125 =216 =343 =512 =729 =1000 =1331 =1728 =2197 =2744 =3375 =4096 =4913 =5832 =6859 =8000 =9261 =10648 =12167 =13824 =15625 =17576 =19683 =21952 =24389 =27000 =29791 =32768 =35937 =39304 =42875 1-100立方表 =46656 =50653 =54872 =59319 =64000 =68921 =74088 =79507 =85184 =91125 =97336 =103823 =110592 =117649 =125000 =132651 =140608 =148877 =157464 =166375 =175616 =185193 =195112 =205379
自然数平方数列和立方数列求和公式
自然数平方数列和立方数列求和公式怎么推导即： (1) 1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)/6 (2) 1^3+2^3+3^3+……+n^3=[n(n+1)/2]^2 推导过程如下： 一. 1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)/6 利用立方差公式 n^3-(n-1)^3=1*[n^2+(n-1)^2+n(n-1)] =n^2+(n-1)^2+n^2-n =2*n^2+(n-1)^2-n 2^3-1^3=2*2^2+1^2-2 3^3-2^3=2*3^2+2^2-3 4^3-3^3=2*4^2+3^2-4 ...... n^3-(n-1)^3=2*n^2+(n-1)^2-n 各等式全相加 n^3-1^3=2*(2^2+3^2+...+n^2)+[1^2+2^2+...+(n-1)^2]-(2+3+4+...+n) n^3-1=2*(1^2+2^2+3^2+...+n^2)-2+[1^2+2^2+...+(n-1)^2+n^2]-n^2-(2+3+4+...+n) n^3-1=3*(1^2+2^2+3^2+...+n^2)-2-n^2-(1+2+3+...+n)+1 n^3-1=3(1^2+2^2+...+n^2)-1-n^2-n(n+1)/2 3(1^2+2^2+...+n^2)=n^3+n^2+n(n+1)/2=(n/2)(2n^2+2n+n+1) =(n/2)(n+1)(2n+1) 故：1^2+2^2+3^2+...+n^2=n(n+1)(2n+1)/6 二. 1^3+2^3+3^3+……+n^3=[n(n+1)/2]^2 证明如下： (n+1)^4-n^4=[(n+1)^2+n^2][(n+1)^2-n^2] =(2n^2+2n+1)(2n+1) =4n^3+6n^2+4n+1
100以内立方表平方表
100以内立方表 13=1 113=1331 213=9261 313=29791 413=68921 23=8 123=1728 223=10648 323=32768 423=74088 33=27 133=2197 233=12167 333=35937 433=79507 43=64 143=2744 243=13824 343=39304 443=85184 53=125 153=3375 253=15625 353=42875 453=91125 63=216 163=4096 263=17576 363=46656 463=97336 73=343 173=4913 273=19683 373=50653 473=103823 83=512 183=5832 283=21952 383=54872 483=110592 93=729 193=6859 293=24389 393=59319 493=117649 103=1000 203=8000 303=27000 403=64000 503=125000 513=132651 613=226981 713=357911 813=531441 913=753571 523=140608 623=238328 723=373248 823=551368 923=778688 533=148877 633=250047 733=389017 833=571787 933=804357 543=157464 643=262144 743=405224 843=592704 943=830584 553=166375 653=274625 753=421875 853=614125 953=857375 563=175616 663=287496 763=438976 863=636056 963=884736 573=185193 673=300763 773=456533 873=658503 973=912673 583=195112 683=314432 783=474552 883=681472 983=941192 593=205379 693=328509 793=493039 893=704969 993=970299 603=216000 703=343000 803=512000 903=729000 1003=1000000 100以内立方表
自然数立方的规律研究
自然数立方的规律研究 我喜欢数学，因为在数学王国里有许多有趣的规律。上学期的一天，我在做正方体体积的计算练习，13=1、23=8、33=27、43=64、53=125……这些答案是否存在什么规律呢？于是我开始仔细地研究。 我把这些答案的各个位数上的数字相加，直到求出的和是个位数时，就发现了一定的规律，于是我列了一张表，如下： 我归纳一下得出这样的普遍规律：自然数n除以3，当余数=1，n3的各个位数上的数字相加，直到求出的和是个位数时，结果得1；当余数=2，n3的各个位数上的数字相加，直到求出的和是个位数时，结果得8；当余数=0，n3的各个位数上的数字相加，直到求出的和是个位数时，结果得9。 这只是偶然吗？后面的自然数立方也遵循这个规律吗？于是我
开始验证我发现的规律。 验证结果让我太高兴了，我立刻把这个发现告诉全家人，大家纷纷拿笔来计算，最后也都符合我发现的这个规律。我太自豪了，这可是我自己动脑筋思考和研究的结果，也许这还是个伟大的发现呢！妈妈笑着提醒我，“你再研究研究，为什么自然数立方会有这样的规律呢？” 对呀，为什么呢？于是，我又进入了新一轮的苦思冥想，经过几番挫折，我都没有成功，后来我逐个突破，先从余数是0的开始，这个自然数n就是3的倍数，即n=3x（x=1，2，3，……），那么， n3=27x3=9×3x3，也就是说这类自然数的立方一定是9的倍数，9的倍数各个位数之和一定是9的倍数，所以将各个位数上的数字相加，
直到求出的和是个位数时，结果一定是9。啊哈，我越来越接近成功了！ 再来看，当余数是1时，这个自然数n就是3的倍数加1，即n=3x+1（x=0，1，2，3，……），那么，n3=（3x+1）3=27x3+27x2+9x+1=9（3x3+3x2+x）+1，也就是说这类自然数的立方一定是9的倍数再加1，那么结果一定是9+1=10，1+0=1，哈哈，第二关闯关成功！ 最后看，当余数是2时，这个自然数n就是3的倍数减1，即n=3x-1（x=1，2，3，……），那么，n3=（3x-1）3=27x3-27x2+9x-1=9（3x3-3x2+x）-1，也就是说这类自然数的立方一定是9的倍数再减1，那么结果一定是9-1=8，哈哈，第三关闯关成功！耶！我兴奋地大叫并跳了起来。 学习数学真是一个快乐的过程，自然数立方的规律问题是我自己在平时学习中发现的，我联系所学的数学知识，仔细思考、归纳总结并想办法证明，让我体会到在数学海洋里遨游的无穷乐趣，我要是能掌握更多的数学知识，我一定会收获更多的快乐。 肖老师留言：下周一上交的是方案，类似于我昨天给你的样本那样简写即可。月底交的文章要详尽，可参考我刚才给你发的范文。
常用自然数平方立方标准表格.doc
常用自然数平方立方 平方立方 2 4 8 3 9 27 4 16 64 5 25 125 6 36 216 7 49 343 8 64 512 9 81 729 10 100 1000 11 121 1331 12 144 1728 13 169 2197 14 196 2744 15 225 3375 16 256 4096 17 289 4913 18 324 5832 19 361 6859 20 400 8000 21 441 9261 22 484 10648 23 529 12167 24 576 13824 25 625 15625 26 676 17576 27 729 19683 28 784 21952 29 841 24389
常用自然数平方立方 平方立方平方立方 2 4 8 16 256 4096 3 9 27 17 289 4913 4 16 64 18 324 5832 5 25 125 19 361 6859 6 36 216 20 400 8000 7 49 343 21 441 9261 8 64 512 22 484 10648 9 81 729 23 529 12167 10 100 1000 24 576 13824 11 121 1331 25 625 15625 12 144 1728 26 676 17576 13 169 2197 27 729 19683 14 196 2744 28 784 21952 15 225 3375 29 841 24389 常用自然数平方立方 平方立方平方立方 2 4 8 16 256 4096 3 9 27 17 289 4913 4 16 64 18 324 5832 5 25 125 19 361 6859 6 36 216 20 400 8000 7 49 343 21 441 9261 8 64 512 22 484 10648 9 81 729 23 529 12167 10 100 1000 24 576 13824 11 121 1331 25 625 15625 12 144 1728 26 676 17576 13 169 2197 27 729 19683 14 196 2744 28 784 21952 15 225 3375 29 841 24389
数字的平方、立方、平方根表
1——50的平方表 1*1=1 2*2=4 3*3=9 4*4=16 5*5=25 6*6=36 7*7=49 8*8=64 9*9=81 10*10=100 11*11=121 12*12=144 13*13=169 14*14=196 15*15=225 16*16=256 17*17=289 18*18=324 19*19=361 20*20=400 21*21=441 22*22=484 23*23=529 24*24=576 25*25=625 26*26=676 27*27=729 28*28=784 29*29=841 30*30=900 31*31=961 32*32=1024 33*33=1089 34*34=1156 35*35=1225 36*36=1296 37*37=1369 38*38=1444 39*39=1521 40*40=1600 41*41=1681 42*42=1764 43*43=1849 44*44=1936 45*45=2025 46*46=2116 47*47=2209 48*48=2304 49*49=2401 50*50=2500 1 —— 20 的立方表 1*1*1=1 2*2*2=8 3*3*3=27 4*4*4=64 5*5*5=125 6*6*6=216 7*7*7=343 8*8*8=512 9*9*9=729 10*10*10=1000 11*11*11=1331 12*12*12=1728 13*13*13=2197 14*14*14=2744 15*15*15=3375 16*16*16=4096 17*17*17=4913 18*18*18=5832 19*19*19=6859 20*20*20=8000 底数为2、3、4、5、6的多次方 底数 2 次 方 3次 方 4次 方 5次 方 6次 方 7次 方 8次 方 9次 方 10次 方 11次 方 12次 方 2 4 8 16 32 64 128 256 512 1024 2048 4096 3 9 27 81 243 729 2187 6561 4 16 64 256 1024 4096 5 25 125 625 3125 6 36 216 1296 7776 46656 0----10的平方根 √0 = 0（表示根号0等于0，下同） √1 = 1√2 = 1.4142135623731 √3 = 1.73205080756888 √4 = 2 √5 = 2.23606797749979 √6 = 2.44948974278318 √7 = 2.64575131106459 √8 = 2.82842712474619 √9 = 3 √10 = 3.16227766016838 1
自然数平方之间的一些规律
自然数平方之间的一些规律 自然数平方之间的一些规律 内容摘要： 1、两个相邻自然数，它们平方数之间有一定的差值，这个差值正好 是这两个相邻自然数之和。 2、我们可以把任意一个被平方数的十位上看作a，个位看作b，那么 它的平方分解的代数式为： （10a＋b）2＝10a×（10a＋2b）＋b2 关键词:自然数平方规律 数学与我们的日常生活息息相关。我对数字（特别是自然数）有着特殊的爱好。我经常留意数字世界，发现它们原来有些有趣的内在规律，下面我就自然数平方之间的一些规律为大家作如下陈述： 一、相邻自然数平方之间的关系 两个相邻自然数，它们平方数之间有一定的差值，这个差值正好是这 两个相邻自然数之和。 如：两个相邻自然数3和4，它们的平方数：32＝9、42＝16，16与9的差是7，7正好是3与4之和。用代数式表示如下： a2－b2＝a＋b（a、b为相邻自然数，a-b=1） 知道了这个规律，我们就可以利用它快速计算出和整十整百数相邻自 然数的平方了。 如：要计算99的平方。想一想：99与100相邻。所以只需用100的平方10000减去99与100之和199，即可得出99的平方了。列式如
下： 992＝1002－（100＋99）＝10000－199＝9801； 如果要计算101的平方，想想，101的平方比100的平方大，所以只需用100的平方10000加上100与101之和201，即得出了101的平 方了。列式如下： 1012＝1002＋（100＋101）＝10000＋201＝10201。 二、两位自然数平方之间的规律 在我们已经熟记了10以内甚至20以内自然数的平方后，我们试图把我们对平方的认识再向上拓展拓展。今天我就两位自然数平方之间的 规律作如下列举说明： 1、十几的平方 112＝10×12＋12 122＝10×14＋22 132＝10×16＋32 142＝10×18＋42 152＝10×20＋52 162＝10×22＋62 172＝10×24＋72
初中常用立方_平方根_立方根表
平方根立方立方根 √0 = 0（表示根号0等于0，下同）√1 = 1 √2 = 1.414 √3 = 1.732 √4 = 2 √5 = 2.236 √6 = 2.449 √7 = 2.646 √8 = 2.828 √9 = 3 √10 = 3.162 √11 = 3.317 √12 = 3.464 √13 = 3.606 √14 = 3.742 √15 = 3.873 √16 = 4 √17 = 4.123 √18 = 4.243 √19 = 4.359 √20 = 4.472 √21 = 4.583 √22 = 4.690 √23 = 4.796 √24 = 4.899 √25 = 5 √26 = 5.099 √27 = 5.196 √28 = 5.292 √29 = 5.385 √30 = 5.477 √31 = 5.568 √32 = 5.657 √33 = 5.745 √34 = 5.831 √35 = 5.916 √36 = 6 √37 = 6.083 √38 = 6.164 √39 = 6.245 √40 = 6.325 √41 = 6.403 √42 = 6.481 √43 = 6.557 √44 = 6.633 √45 = 6.708 √46 = 6.782 √47 = 6.856 √48 = 6.928 √49 = 7 √50 = 7.071 √51 = 7.141 √52 = 7.211 √53 = 7.280 √54 = 7.348 √55 = 7.416 √56 = 7.483 √57 = 7.550 √58 = 7.616 √59 = 7.681 √60 = 7.746 √61 = 7.810 √62 = 7.874 √63 = 7.937 √64 = 8 √65 = 8.062 √66 = 8.124 √67 = 8.185 √68 = 8.246 √69 = 8.307 √70 = 8.367 √71 = 8.426 √72 = 8.485 √73 = 8.544 √74 = 8.602 √75 = 8.660 √76 = 8.718 √77 = 8.775 √78 = 8.832 √79 = 8.888 √80 = 8.944 √81 = 9 √82 = 9.055 √83 = 9.110 √84 = 9.165 √85 = 9.220 √86 = 9.274 √87 = 9.327 √88 = 9.381 √89 = 9.434 √90 = 9.487 √91 = 9.539 √92 = 9.592 √93 = 9.644 √94 = 9.695 √95 = 9.747 √96 = 9.798 √97 = 9.849 √98 = 9.900 √99 = 9.950 √100 = 10 1^3=1 2^3=8 3^3=27 4^3=64 5^3=125 6^3=216 7^3=343 8^3=512 9^3=729 10^3=1000 11^3=1331 12^3=1728 13^3=2197 14^3=2744 15^3=3375 16^3=4096 17^3=4913 18^3=5832 19^3=6859 20^3=8000 21^3=9261 22^3=10648 23^3=12167 24^3=13824 25^3=15625 26^3=17576 27^3=19683 28^3=21952 29^3=24389 30^3=27000 31^3=29791 32^3=32768 33^3=35937 34^3=39304 35^3=42875 36^3=46656 37^3=50653 38^3=54872 39^3=59319 40^3=64000 41^3=68921 42^3=74088 43^3=79507 44^3=85184 45^3=91125 46^3=97336 47^3=103823 48^3=110592 49^3=117649 50^3=125000 51^3=132651 52^3=140608 53^3=148877 54^3=157464 55^3=166375 56^3=175616 57^3=185193 58^3=195112 59^3=205379 60^3=216000 61^3=226981 62^3=238328 63^3=250047 64^3=262144 65^3=274625 66^3=287496 67^3=300763 68^3=314432 69^3=328509 70^3=343000 71^3=357911 72^3=373248 73^3=389017 74^3=405224 75^3=421875 76^3=438976 77^3=456533 78^3=474552 79^3=493039 80^3=512000 81^3=531441 82^3=551368 83^3=571787 84^3=592704 85^3=614125 86^3=636056 87^3=658503 88^3=681472 89^3=704969 90^3=729000 91^3=753571 92^3=778688 93^3=804357 94^3=830584 95^3=857375 96^3=884736 97^3=912673 98^3=941192 99^3=970299 100^3=1000000 3√0 = 0 3√1 = 1 3√2 = 1.260 3√3 = 1.442 3√4 = 1.587 3√5 = 1.710 3√6 = 1.817 3√7 = 1.913 3√8 = 2 3√9 = 2.080 3√10 = 2.154 3√11 = 2.224 3√12 = 2.289 3√13 = 2.351 3√14 = 2.410 3√15 = 2.466 3√16 = 2.520 3√17 = 2.571 3√18 = 2.621 3√19 = 2.668 3√20 = 2.714 3√21 = 2.759 3√22 = 2.802 3√23 = 2.844 3√24 = 2.884 3√25 = 2.924 3√26 = 2.962 3√27 = 3 3√28 = 3.037 3√29 = 3.072 3√30 = 3.107 3√31 = 3.141 3√32 = 3.175 3√33 = 3.206 3√34 = 3.240 3√35 = 3.271 3√36 = 3.302 3√37 = 3.332 3√38 = 3.362 3√39 = 3.391 3√40 = 3.420 3√41 = 3.448 3√42 = 3.476 3√43 = 3.503 3√44 = 3.530 3√45 = 3.557 3√46 = 3.583 3√47 = 3.609 3√48 = 3.634 3√49 = 3.659 3√50 = 3.684 3√51 = 3.708 3√52 = 3.733 3√53 = 3.756 3√54 = 3.780 3√55 = 3.803 3√56 = 3.826 3√57 = 3.849 3√58 = 3.871 3√59 = 3.893 3√60 = 3.915 3√61 = 3.936 3√62 = 3.958 3√63 = 3.979 3√64 = 4 3√65 = 4.021 3√66 = 4.041 3√67 = 4.062 3√68 = 4.082 3√69 = 4.102 3√70 = 4.121 3√71 = 4.141 3√72 = 4.160 3√73 = 4.179 3√74 = 4.198 3√75 = 4.217 3√76 = 4.236 3√77 = 4.254 3√78 = 4.273 3√79 = 4.291 3√80 = 4.309 3√81 = 4.327 3√82 = 4.344 3√83 = 4.362 3√84 = 4.380 3√85 = 4.397 3√86 = 4.414 3√87 = 4.431 3√88 = 4.448 3√89 = 4.465 3√90 = 4.481 3√91 = 4.498 3√92 = 4.514 3√93 = 4.531 3√94 = 4.547 3√95 = 4.563 3√96 = 4.579 3√97 = 4.595 3√98 = 4.610 3√99 = 4.626 3√100 = 4.642
自然数的平方和立方的一些规律及其证明
平方和公式：1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)/6 立方和公式：1^3+2^3+...+n^3=[n(n+1)/2]^2 首先给出网上的推导： 1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)/6 利用立方差公式 n^3-(n-1)^3=1*[n^2+(n-1)^2+n(n-1)] =n^2+(n-1)^2+n^2-n =2*n^2+(n-1)^2-n 2^3-1^3=2*2^2+1^2-2 3^3-2^3=2*3^2+2^2-3 4^3-3^3=2*4^2+3^2-4 ...... n^3-(n-1)^3=2*n^2+(n-1)^2-n 各等式全相加 n^3-1^3=2*(2^2+3^2+...+n^2)+[1^2+2^2+...+(n-1)^2]-(2+3+4+...+n) n^3-1=2*(1^2+2^2+3^2+...+n^2)-2+[1^2+2^2+...+(n-1)^2+n^2]-n^2-(2+3+4+...+n) n^3-1=3*(1^2+2^2+3^2+...+n^2)-2-n^2-(1+2+3+...+n)+1 n^3-1=3(1^2+2^2+...+n^2)-1-n^2-n(n+1)/2 3(1^2+2^2+...+n^2)=n^3+n^2+n(n+1)/2=(n/2)(2n^2+2n+n+1)=(n/2)(n+1)(2n+1) 1^2+2^2+3^2+...+n^2=n(n+1)(2n+1)/6 1^3+2^3+3^3+……+n^3=[n(n+1)/2]^2 (n+1)^4-n^4=[(n+1)^2+n^2][(n+1)^2-n^2] =(2n^2+2n+1)(2n+1) =4n^3+6n^2+4n+1 2^4-1^4=4*1^3+6*1^2+4*1+1 3^4-2^4=4*2^3+6*2^2+4*2+1 4^4-3^4=4*3^3+6*3^2+4*3+1 ...... (n+1)^4-n^4=4*n^3+6*n^2+4*n+1 各式相加有 (n+1)^4-1=4*(1^3+2^3+3^3...+n^3)+6*(1^2+2^2+...+n^2)+4*(1+2+3+...+n)+n 4*(1^3+2^3+3^3+...+n^3)=(n+1)^4-1+6*[n(n+1)(2n+1)/6]+4*[(1+n)n/2]+n =[n(n+1)]^2 1^3+2^3+...+n^3=[n(n+1)/2]^2 平方和的经典题目： 立方和的另类推导： (1)
常用平方、立方标准表格.doc
1—— 100 的平方表 =1296 =5184 =1 =1369 =5329 =4 =1444 =5476 =9 =1521 =5625 =16 =1600 =5776 =25 =1681 =5929 =36 =1764 =6084 =49 =1849 =6241 =64 =1936 =6400 =81 =2025 =6561 =100 =2116 =6724 =121 =2209 =6889 =144 =2304 =7056 =169 =2401 =7225 =196 =2500 =7396 =225 =2601 =7569 =256 =2704 =7744 =289 =2809 =7921 =324 =2916 =8100 =361 =3025 =8281 =400 =3136 =8464 2 =441 =3249 =8649 =484 =3364 =8836 =529 =3481 =9025 =576 =3600 =9216 =625 =3721 =9409 =676 =3844 =9604 =729 =3969 =9801 =784 =4096 =10000 =841 =4225 =900 =4356 =961 =4489 =1024 =4624 =1089 =4761 =1156 =4900 =1225 =5041=1
=8 =50653 =373248 =27 =54872 =389017 =64 =59319 =405224 =125 =64000 =421875 =216 =68921 =438976 =343 =74088 =456533 =512 =79507 =474552 =729 =85184 =493039 =1000 =91125 =512000 =1331 =97336 =531441 =1728 =103823 =551368 =2197 =110592 =571787 =2744 =117649 =592704 =3375 =125000 =614125 =4096 =132651 =636056 =4913 =140608 =658503 =5832 =148877 =681472 =6859 =157464 =704969 =8000 =166375 =729000 =9261 =175616 =753571 =10648 =185193 2=778688 =12167 =195112 =804357 =13824 =205379 =830584 =15625 =216000 =857375 =17576 =226981 =884736 =19683 =238328 =912673 =21952 =250047 =941192 =24389 =262144 =970299 =27000 =274625 =1000000 =29791 =287496 =32768 =300763 =35937 =314432 =39304 =328509 =42875 =343000 1-100 立方表 =46656=357911
自然数平方数列和立方数列求和公式
自然数平方数列和立方数列求和公式怎么推导？ 即： (1) 1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)/6 (2) 1^3+2^3+3^3+……+n^3=[n(n+1)/2]^2 推导过程如下： 一. 1^2+2^2+3^2+……+n^2=n(n+1)(2n+1)/6 利用立方差公式 n^3-(n-1)^3=1*[n^2+(n-1)^2+n(n-1)] =n^2+(n-1)^2+n^2-n =2*n^2+(n-1)^2-n 2^3-1^3=2*2^2+1^2-2 3^3-2^3=2*3^2+2^2-3 4^3-3^3=2*4^2+3^2-4 ...... n^3-(n-1)^3=2*n^2+(n-1)^2-n 各等式全相加 n^3-1^3=2*(2^2+3^2+...+n^2)+[1^2+2^2+...+(n-1)^2]-(2+3+4+...+n) n^3-1=2*(1^2+2^2+3^2+...+n^2)-2+[1^2+2^2+...+(n-1)^2+n^2]-n^2-(2+3+4+...+n)
n^3-1=3*(1^2+2^2+3^2+...+n^2)-2-n^2-(1+2+3+...+n)+1 n^3-1=3(1^2+2^2+...+n^2)-1-n^2-n(n+1)/2 3(1^2+2^2+...+n^2)=n^3+n^2+n(n+1)/2=(n/2)(2n^2+2n+n+1) =(n/2)(n+1)(2n+1) 故：1^2+2^2+3^2+...+n^2=n(n+1)(2n+1)/6 二. 1^3+2^3+3^3+……+n^3=[n(n+1)/2]^2 证明如下： (n+1)^4-n^4=[(n+1)^2+n^2][(n+1)^2-n^2] =(2n^2+2n+1)(2n+1) =4n^3+6n^2+4n+1 2^4-1^4=4*1^3+6*1^2+4*1+1