Question

已知数列{an}满足：a1=2/3,且an+1=1/3an+2×(1/3)^n+1求数列{an}的

已知数列{an}满足：a1=2/3,且an+1=1/3an+2×(1/3)^n+1求数列{an}的前n项和Sn

Answer 1

a1=2/3, an+1=1/3an+2×(1/3)^n+1

方程两边同除以(1/3)^n+1 得
[an+1/(1/3)^n+1]-an/(1/3)^n=2
{an/(1/3)^n}成等差
an/(1/3)^n=3a1+2(n-1)=2n
an=2n*(1/3)^n
Sn=2*1/3+4*(1/3)^2+6*(1/3)^3+...+(2n-2)(1/3)^(n-1)+2n*(1/3)^n
1/3Sn= 2*(1/3)^2+4*(1/3)^3+...+(2n-4)(1/3)^(n-1)+2n*(1/3)^(n+1)
两式相减得
2/3Sn=2[1/3+(1/3)^2+(1/3)^3+....+(1/3)^n]-2n*(1/3)^(n+1)=2[1/3(1-(1/3)^n]/2/3-2n*(1/3)^(n+1)
2/3Sn=1-(1/3)^n-2n*(1/3)^(n+1)
Sn=3/2+(2n-3)/2*(1/3)^n