Question

已知数列{an}满足a1=1,an=a1+2a2+3a3....+(n-1)an-1 (n>=2),则{an}的通项是什么

要过程，详细。用高一知识解

Answer 1

a(n+1)=a(1)+2a(2)+...+na(n), a(2) = a(1) = 1.
a(n+2)=a(1)+2a(2)+...+na(n)+(n+1)a(n+1) = a(n+1) + (n+1)a(n+1) = (n+2)a(n+1),
a(n+2)/(n+2)! = a(n+1)/(n+1)!,
{a(n+1)/(n+1)!}是首项为a(2)/2 = 1/2,的常数数列。

a(n+1)/(n+1)! = 1/2,
a(n+1) = (n+1)!/2,

{a(n)}的通项为，
a(1)=1,
n>=2时，a(n) = n!/2.

Answer 2

a(1)=1

a(n)=n!/2