正项数列an满足：a1=3/2,a（n+1）=3an/2an+3

题目

正项数列an满足：a1=3/2,a（n+1）=3an/2an+3
数列bn满足bn·an=3（1-1/2^n),求bn的前n和

答案

a(n+1)=3an/(2an +3)
1/a(n+1)=(2an +3)/(3an)=1/an +2/3
1/a(n+1)-1/an=2/3,为定值
1/a1=1/(3/2)=2/3,数列{1/an}是以2/3为首项,2/3为公差的等差数列
1/an=2/3+ (2/3)(n-1)=2n/3
bn·an=3(1- 1/2ⁿ)
bn=3(1- 1/2ⁿ)·(1/an)
=3(1- 1/2ⁿ)·(2n/3)
=2n - n/2^(n-1)
Tn=b1+b2+...+bn=2(1+2+...+n)- [1/1+2/2+3/2²+...+n/2^(n-1)]
令Bn=1/1+2/2+3/2²+...+n/2^(n-1)
则Bn/2=1/2+2/2²+...+(n-1)/2^(n-1)+ n/2ⁿ
Bn-Bn/2=Bn/2
=1+1/2+1/2²+...+1/2^(n-1) -n/2ⁿ
=1×(1-1/2ⁿ)/(1-1/2) -n/2ⁿ
=2-(n+2)/2ⁿ
Bn=4- (n+2)/2^(n-1)
Tn=2(1+2+...+n)-Bn
=2n(n+1)/2 -4 +(n+2)/2^(n-1)
=n²+n +(n+2)/2^(n-1) -4