Question

记sn为等比数列an的前n项和,a1=1/3

一：已知等比数列｛an｝中,a1=1/3,公比q=1/3、1：Sn为｛an｝的前n项和,证明Sn=（1-an）/2.2：设bn=log以3为底a1的对数+log以3为底a2的对数+…+log以3为底an的对数,求数列｛bn｝的通项.

Answer 1

（1）a1=1/3,公比q=1/3
所以：an=1/(3^n)
Sn=a1(1-q^n)/(1-q) =(1/3)(1-1/3^n)/(2/3)= (1-1/3^n)/2 =(1-an)/2
(2) bn=log(3)a1+log(3)a2+…+log(3)an
=log(3)[a1×a2×…×an]
=log(3)[1/3×1/(3^2) ×…×1/(3^n) ]
=log(3)[1/3^(1+2+…+n) ]
=－log(3)[3^(1+2+…+n) ]
=－(1+2+…+n)
=－n(1+n) /2