已知数列{an}满足a1=1,an=a1+2a2+3a3+…+(n-1)an-1,则n≥2时,数列{an}的通项an=(  ) A.n!2 B.(n+1)!2 C.n! D.(n+1)!

问题描述:

已知数列{an}满足a1=1,an=a1+2a2+3a3+…+(n-1)an-1,则n≥2时,数列{an}的通项an=(  )
A.

n!
2

B.
(n+1)!
2

C. n!
D. (n+1)!

由an=a1+2a2+3a3+…+(n-1)an-1(n≥2),得nan+an=a1+2a2+3a3+…+(n-1)an-1+nan(n≥2),∴(n+1)•an=an+1(n≥2),则an+1an=n+1(n≥2),又a1=1,∴a2=1,∴a3a2=3,a4a3=4,…,anan−1=n.累积得an=n!2...