高中數學數列

a(n+1) = a(n) + (3/n)a(n) = [(n+3)/n] a(n)
= [(n+3)/n] [(n+2)/(n-1)] a(n-1)
= [ (n+3)/n] [(n+2)/(n-1)] ... (4/1) a(1)
= [(n+3)(n+2)(n+1)/(3．2．1)] a(1)
= (n+3)(n+2)(n+1)/3


2015-06-15 03:58:05 補充：
所以
a(n) = (n+2)(n+1)n/3.

檢查:
a(1) = 2
a(n+1)-a(n) = (n+3)(n+2)(n+1)/3 - (n+2)(n+1)n/3 = (n+2)(n+1)
所以 n(a(n+1)-a(n)) = (n+2)(n+1)n = 3a(n).