设Sn等比数列{an}的前n项和,且a2=1/9,S2=4/9 ,设bn=n/an,求数列{bn}的前n项和Tn

问题描述:

设Sn等比数列{an}的前n项和,且a2=1/9,S2=4/9 ,设bn=n/an,求数列{bn}的前n项和Tn

已知a2=1/9,
S2=a1+a2=4/9
a1=4/9-1/9=1/3
公比q=a2/a1=4/3
所以an=(1/3)*(4/3)^(n-1)
bn=3n*(3/4)^(n-1)
Tn=3[1+2*(3/4)+3*(3/4)^2+.+n*(3/4)^(n-1)]
(3/4)Tn=3[3/4+2*(3/4)^2+3*(3/4)^3+.+n*(3/4)^n]
Tn-(3/4)Tn=3[1+(3/4)+(3/4)^2+.+(3/4)^(n-1)-n*(3/4)^n]
(1/4)Tn=3{[1-(3/4)^n]/(1-3/4)-n*(3/4)^n}
Tn=12{4[1-(3/4)^n-n*(3/4)^n}
=12[4-(n+4)*(3/4)^n]