已知等比数列｛an},首项为81,数列{bn}满足bn=log3为底an,其前n项和Sn1)证明｛bn}为等差数列2）若S11≠S12,且S11最大,求{bn}的公差d的范围
问题描述：
已知等比数列｛an},首项为81,数列{bn}满足bn=log3为底an,其前n项和Sn
1)证明｛bn}为等差数列
2）若S11≠S12,且S11最大,求{bn}的公差d的范围
答
an=81q^(n-1)
bn=log3(an)
=log3[81q^(n-1)
=log3(81)+(n-1)log3(q)
=4+(n-1)log3(q)
bn-bn-1=4+(n-1)log3(q)-4-(n-2)log3(q)
=log3(q)
为定值。{bn}为等差数列。
b1=4，公差为log3(q)
S11>0
S11=4*11+55log3(q)>0
log3(q)>-4/5
S12S12=4*12+66log3(q)log3(q)公差的取值范围为(-4/5,-24/33)
答
(1)等比数列｛an},首项为81设an=a1*q^(n-1)=81*q^(n-1)数列{bn}满足bn=log3为底an∴bn=log3为底[81*q^(n-1)]=log3为底81+log3为底q^(n-1)=4+(n-1)log3为底q=log3为底q*n+4-log3为底q∵log3为底q为常数,∴bn为以4为首...