对于正项数列{an},定义H=n/(a1+2a2+3a3+.+nan)为an的“给力值”,现知数列{an}的“给力值”为H=1/n,则数列{an}的通项公式为?

Question

对于正项数列{an},定义H=n/(a1+2a2+3a3+.+nan)为an的“给力值”,
现知数列{an}的“给力值”为H=1/n,则数列{an}的通项公式为?

Answer 1

n=1时,1/1 = 1/a(1),a(1)=1.
1/n = n/[a(1)+2a(2)+...+na(n)],
a(1)+2a(2)+...+na(n)=n^2,
a(1)+2a(2)+...+na(n)+(n+1)a(n+1)=(n+1)^2,
2n+1 = (n+1)^2 - n^2 = (n+1)a(n+1),
a(n+1) = (2n+1)/(n+1) = [2(n+1)-1]/(n+1),
a(n) = (2n-1)/n