Question

求和W=

C

0n

+4

C

1n

+7

C

2n

+10

C

3n

+…+(3n+1)

C

nn

．

Answer 1

∵a_n=3n+1为等差数列，∴a₀+a_n=a₁+a_n-1=…，
而

C

kn

=

C

n-kn

，（运用反序求和方法），
∵W=

C

0n

+4

C

1n

+7

C

2n

+…+(3n-2)

C

n-1n

+(3n+1)

C

nn

①，
=(3n+1)

C

nn

+(3n-2)

C

n-1n

+(3n-5)

C

n-2n

+…+4

C

1n

+

C

0n

∴W=(3n+1)

C

0n

+(3n-2)

C

1n

+(3n-5)

C

n-2n

+…+4

C

1n

+

C

0n

②，
①+②得2W=(3n+2)(

C

0n

+

C

1n

+

C

2n

+…+

C

nn

)=(3n+2)×2n，
∴W=（3n+2）×2^n-1．