## [國中]連老師都算不出來的奇怪數學問題

1234door

### Re: [國中]連老師都算不出來的奇怪數學問題

1234door 寫到:

Hint:
S1 = 1 + 2 + 3 + .. + n = n(n+1)/2
S2 = 12+ 22 + .. + n2 = n(n+1)(2n+1)/6 = n(n+1)/2 * [(2n+1)/3] = S1(2n+1)/3
S3 = 13+ 23 + .. + n3 = [(n+1)4-(n+1)]/4 -3S2/2 -S1 = n(n+1)(n2+3n+3)/4-3S2/2 -S1 = S1(n2+3n+3)/2-3S2/2 -S1 = S1的倍數
...
Sm = [(n+1)m+1-(n+1)] / (m+1) + S1的倍數

lskuo

### Re: [國中]連老師都算不出來的奇怪數學問題

lskuo 寫到:
1234door 寫到:

Hint:
S1 = 1 + 2 + 3 + .. + n = n(n+1)/2
S2 = 12+ 22 + .. + n2 = n(n+1)(2n+1)/6 = n(n+1)/2 * [(2n+1)/3] = S1(2n+1)/3
S3 = 13+ 23 + .. + n3 = [(n+1)4-(n+1)]/4 -3S2/2 -S1 = n(n+1)(n2+3n+3)/4-3S2/2 -S1 = S1(n2+3n+3)/2-3S2/2 -S1 = S1的倍數
...
Sm = [(n+1)m+1-(n+1)] / (m+1) + S1的倍數

S3 = 13+ 23 + .. + n3 = [(n+1)4-(n+1)]/4 -3S2/2 -S1 = n(n+1)(n2+3n+3)/4-3S2/2 -S1 = S1(n2+3n+3)/2-3S2/2 -S1 = S1的倍數

1234door

### Re: [國中]連老師都算不出來的奇怪數學問題

(n+1)m = nm + C(m,1)nm-1 + C(m,2)nm-2 + ... + C(m,m-1)n + 1

C(m,k) = m!/k!/(m-k)!

1m = 1
2m = 1m + C(m,1)2m-1 + C(m,2)2m-2 + ... + C(m,m-1)2 + 1
...
(n+1)m = nm + C(m,1)nm-1 + C(m,2)nm-2 + ... + C(m,m-1)n + 1

(n+1)m -1 = C(m,1)Sm-1 + C(m,2)Sm-2 + ... + C(m,m-1)S1 + n

C(m,1)Sm-1 = (n+1)m -(n+1) - C(m,2)Sm-2 - ... - C(m,m-1)S1

C(4,1) = 4, C(4,2)=6, C(4,3)=4

4S3 = (n+1)4 -(n+1) -6S2  - 4S1

lskuo