等比数列的各项均为正数且2a1+3a2=1a3^2=9a2a6

a(n)=aq^(n-1),a>0,q>0.
1 = 2a(1)+3a(2)=2a+3aq,
9a(2)a(6)=9a^2q^6 = [a(3)]^2 = a^2q^4,9q^2 = 1,q = 1/3.
1 = 2a+3aq = a(2+3q)=a(2+3/3)=3a,a = 1/3.
a(n) = (1/3)(1/3)^(n-1) = (1/3)^n = 3^(-n)
log_{3}[a(n)] = log_{3}[3^(-n)] = -n,
b(n) = log_{3}[a(1)] + log_{3}[a(2)] + ...+ log_{3}[a(n)]
= -n(n+1)/2,
1/b(n) = -2/[n(n+1)] = -2/n + 2/(n+1),
1/b(1)+1/b(2)+...+1/b(n-1)+1/b(n) = -2/1 + 2/2 - 2/2 + 2/3 - ...- 2/(n-1) + 2/n - 2/n + 2/(n+1)
= -2/1 + 2/(n+1)
= -2n/(n+1)