已知等比数列{an},首项为81,数列{bn}=log3an,其前n项和为Sn、证明{bn}是等差数

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/05/18 02:16:13

证：
设等比数列{an}公比为q,对于数列{bn},对数有意义,q>0
an=a1×q^(n-1)
n=1时,b1=log3(a1)=log3(81)=4
n≥2时,
bn=log3(an)
=log3(a1×q^(n-1))
=log3(a1)+log3(q^(n-1))
=log3(81)+(n-1)log3(q)
=(n-1)log3(q) +4
b(n-1)=log3(a(n-1))
=log3(a1×q^(n-2))
=(n-2)log3(q) +4
bn-b(n-1)=(n-1)log3(q) +4-(n-2)log3(q) -4=log3(q),为定值.
数列{bn}是以4为首项,log3(q)为公差的等差数列.

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