- !

" ", "" " ". .
18.03.2013

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, , n .


, nN :

1) 1 + 2 + 3 + ... + n = n(n + 1)

2

;
2) 12 + 22 + 32 + ... + n2 = n(n + 1)(2n + 1)

6

;
3) 13 + 23 + 33 + ... + n3 = n2(n + 1)2

4

;



1 + 2 + 3 + ... + n = n(n + 1)

2

, . , , . , 1 + 2 + 3 + ... + n, :

1 + 2 + ... + n - 1 + n
n + n - 1 + ... + 2 + 1

, , , , .. 2(1 + 2 + 3 + ... + n). , , , , n + 1 ( ). () n + 1, 1, 1 . n + 1. n ( ), : 2(1 + 2 + 3 + ... + n) = n(n + 1), .



12 + 22 + 32 + ... + n2 = n(n + 1)(2n + 1)

6

.

.

, n = 1.

12 = 123/6 = 1.

.

.

kN

12 + 22 + 32 + ... + k2 = k(k + 1)(2k + 1)

6

;

, k + 1, ..,

12 + 22 + 32 + ... + k2 + (k + 1)2 =? (k + 1)(k + 2)(2k + 3)

6

; (*)

,

k(k + 1)(2k + 1)

6

+ (k + 1)2 =? (k + 1)(k + 2)(2k + 3)

6

;

6, k + 1:

k(2k + 1) + 6(k + 1) =? (k + 2)(2k + 3);

2k2 + 7k + 6 = 2k2 + 7k + 6.

, , , (*) - . , .



13 + 23 + 33 + ... + n3 = n2(n + 1)2

4

.

.

, n = 1.

13 = 1222/4 = 1.

.

.

kN

3) 13 + 23 + 33 + ... + k3 = k2(k + 1)2

4

;

, k + 1, ..,

3) 13 + 23 + 33 + ... + (k + k + 1)3 =? (k + 1)2(k + 2)2

4

; (**)

,

k2(k + 1)2

4

+ (k + 1)3 =? (k + 1)2(k + 2)2

4

;

4, (k + 1)2:

k2 + 4(k + 1) =? (k + 2)2;

k2 + 4k + 4 = k2 + 4k + 4;

, , , (**) - . , .



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