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ORIENTATION GÉNÉRALE  - M'écrire - Édition du: 26/09/2012

 

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Puissance n

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SOMMES de PUISSANCES

successives d'un même nombre

 

Table

 

 

Mise en bouche –

Somme des puissances d'un nombre = puissance

 

*    Trois motifs sympathiques
 

  30 + 31  + 32 + 33 + 34

= 1 + 3 + 9 + 2 7 + 81

= 121

= 11²

  70 + 71  + 72 + 73

= 1 + 7 + 49 + 343

= 400

= 20²

180 + 181  + 182

= 1 + 18 + 324

= 343

=   73

 

Exploration jusqu'à puissance = 10 000 et un cumul des puissances jusqu'à 100

 

*    Une curiosité

 

     70  +   71  +   72

+ 180 + 181  + 182

= 1 + 7 + 49 + 343

= 400

= 20²

 

Voir Pépites

 

 

 

Table des SOMMES DES PUISSANCES successives d'un même nombre

par ordre croissant de la somme

 

 

Exemples de lecture de la table

2   1   3

 

signifie

20 + 21 = 3

2   3   15

 

signifie

20 + 21 + 23 = 15

4   4   341

 

signifie

40 + 41 + 42 + 43 + 44 = 341

 

Table

N              q                                         SP

 

2               1                                             3  

3               1                                             4  

4               1                                             5  

5               1                                             6  

2               2                                             7  

6               1                                             7  

7               1                                             8  

8               1                                             9  

9               1                                           10  

10             1                                           11  

11             1                                           12  

3               2                                           13  

12             1                                           13  

2               3                                           15  

4               2                                           21  

2               4                                           31  

5               2                                           31  

3               3                                           40  

6               2                                           43  

7               2                                           57  

2               5                                           63  

8               2                                           73  

4               3                                           85  

9               2                                           91  

10             2                                         111  

3               4                                         121  

2               6                                         127  

11             2                                         133  

5               3                                         156  

12             2                                         157  

2               7                                         255  

6               3                                         259  

4               4                                         341  

3               5                                         364  

7               3                                         400  

2               8                                         511  

8               3                                         585  

5               4                                         781  

9               3                                         820  

2               9                                      1 023  

3               6                                      1 093  

10             3                                      1 111  

4               5                                      1 365  

11             3                                      1 464  

6               4                                      1 555  

12             3                                      1 885  

2               10                                     2 047  

7               4                                      2 801  

3               7                                      3 280  

5               5                                      3 906  

8               4                                      4 681  

4               6                                      5 461  

9               4                                      7 381   

6               5                                      9 331  

3               8                                      9 841  

10             4                                     11 111  

11             4                                     16 105  

5               6                                     19 531  

7               5                                     19 608  

4               7                                     21 845  

12             4                                     22 621  

3               9                                     29 524  

8               5                                     37 449  

6               6                                     55 987  

9               5                                     66 430  

4               8                                     87 381  

3               10                                    88 573  

5               7                                     97 656  

10             5                                   111 111  

7               6                                   137 257  

11             5                                   177 156  

12             5                                   271 453  

8               6                                   299 593  

6               7                                   335 923  

4               9                                   349 525  

5               8                                   488 281  

9               6                                   597 871  

7               7                                   960 800  

10             6                                1 111 111  

4               10                               1 398 101  

11             6                                1 948 717  

6               8                                2 015 539  

8               7                                2 396 745  

5               9                                2 441 406  

12             6                                3 257 437  

9               7                                5 380 840  

7               8                                6 725 601  

10             7                              11 111 111  

6               9                              12 093 235  

5               10                             12 207 031  

8               8                              19 173 961  

11             7                              21 435 888  

12             7                              39 089 245  

7               9                              47 079 208  

9               8                              48 427 561  

6               10                             72 559 411  

10             8                             111 111 111  

8               9                             153 391 689  

11             8                             235 794 769  

7               10                            329 554 457  

9               9                             435 848 050  

12             8                             469 070 941  

10             9                          1 111 111 111  

8               10                         1 227 133 513  

11             9                          2 593 742 460  

9               10                         3 922 632 451  

12             9                          5 628 851 293  

10             10                       11 111 111 111  

11             10                       28 531 167 061  

12             10                       67 546 215 517  

 

 

 

 

SOMMES Nombres successifs et puissance

 

Exemples de lecture de la table

2   2   5

 

signifie

12 + 22 = 5

2   3   14

 

signifie

12 + 22 + 32 = 14

4   3   98

 

signifie

14 + 24 + 34  = 98

Avec le carré (2), il faut deux nombres consécutifs (2) pour avoir une somme de 5.

 

Table

n              q                  SP

 

2              2                  5

3              2                  9

2              3                  14

4              2                  17

2              4                  30

5              2                  33

3              3                  36

2              5                  55

6              2                  65

2              6                  91

4              3                  98

3              4                  100

7              2                  129

2              7                  140

2              8                  204

3              5                  225

8              2                  257

5              3                  276

2              9                  285

4              4                  354

2              10                 385

3              6                  441

2              11                 506

9              2                  513

2              12                 650

3              7                  784

6              3                  794

4              5                  979

10            2                  1025

3              8                  1296

5              4                  1300

3              9                  2025

11            2                  2049

4              6                  2275

7              3                  2316

3              10                 3025

12            2                  4097

3              11                 4356

5              5                  4425

4              7                  4676

6              4                  4890

3              12                 6084

8              3                  6818

4              8                  8772

5              6                  12201

4              9                  15333

7              4                  18700

9              3                  20196

6              5                  20515

4              10                 25333

5              7                  29008

4              11                 39974

10            3                  60074

4              12                 60710

5              8                  61776

6              6                  67171

8              4                  72354

7              5                  96825

5              9                  120825

11            3                  179196

6              7                  184820

5              10                 220825

9              4                  282340

7              6                  376761

5              11                 381876

6              8                  446964

8              5                  462979

12            3                  535538

5              12                 630708

6              9                  978405

 

 

Suite en Table des Sommes de puissances successives d'un même nombre

 

 


 

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