Question

已知数列an中a1+a2……an=（3^n-2^）/2^n 求证an是等比数列

Answer 1

a1+a2……+an=（3^n-2^n）/2^n

即Sn=（3^n-2^n）/2^n,

Sn=(3/2)^n-1.

当n=1时,a1=S1=3/2-1=1/2.

当n≥2时,an= Sn-S(n-1)= (3/2)^n-1-[(3/2)^(n-1)-1]

= (3/2)^n-(3/2)^(n-1)= 1/2•(3/2)^(n-1),

∴an=1/2•(3/2)^(n-1).(n∈N*)

数列{an}是公比为3/2的等比数列.