F X ( )  (1  X ) N  1  (1  X C )  N 0  XC N 1  X C 2 N 2  X C 3 N 3 

1) Xét hàm số: f x ^{( )}  ⁽¹  x ^{) n  1}  ⁽¹  x C ⁾  n ⁰  xC n ¹  x C ² n ²  x C ³ n ³  ^...  x C ⁿ n ⁿ 

 (1  x C ) _n ⁰  ( x  x C ² ) ¹ _n  ( x ²  x C ³ ) _n ²  ( x ³  x C ⁴ ) _n ³  ... (  x ⁿ  x ^{n  1} ) C _n ⁿ

Ta có: f x ( )  (1  x ) ^{n  1}  f x '( )  ( n  1)(1  x ) ⁿ  f "( ) x  ( n  1) (1 n  x ) ^{n  1}  f "(1)  ( n  1) .2 n ^{n  1}

hay f x ( )  (1  x C ) _n ⁰  ( x  x C ² ) ¹ _n  ( x ²  x C ³ ) _n ²  ( x ³  x C ⁴ ) _n ³  ... (  x ⁿ  x ^{n  1} ) C _n ⁿ

0 1 2 2 2 3 3 1



n n n

'( ) (1 2 ) (2 3 ) (3 4 ) ... ( ( 1) )

f x C x C x x C x x C nx n x C

           

n n n n n

1 2 2 3 2 1

 

          

"( ) 2 (2 6 ) (6 12 ) ... ( ( 1) ( 1) )

f x C x C x x C n n x n nx C

n n n n

n

1 2 2 2 3 2

     

f C C C n C

"(1) 2 2.2 2.3 ... 2

1 2 2 2 3 2 1 1 2 2 2 3 2 2

2 C _n 2.2 C _n 2.3 C _n ... 2 n C _n ⁿ ( n 1) .2 n ^{n } C _n 2 C _n 3 C _n ... n C _n ⁿ ( n 1) .2 n ^{n }

             

Khi n  2012 , ta có: 1 ² C ₂₀₁₂ ¹  2 ² C ₂₀₁₂ ²  3 ² C ₂₀₁₂ ³  ... 2012  ² C ₂₀₁₂ ²⁰¹²  2013.2012.2 ²⁰¹⁰ =S