Question

已知数列an满足a1+2a2+2^2a3+...+2^n-1an=2^2n-1(n∈N*)

Answer 1

1.

a1+2a2+2²a3+...+[2^(n-1)]an=2^(2n-1) (1)

a1+2a2+2²a3+...+[2^(n-2)]a(n-1)=2^(2n-3) (2)

(1)-(2)

[2^(n-1)]an=2^(2n-1)-2^(2n-3)=3×2^(2n-3)

an=3×2^(2n-3)/2^(n-1)=3×2^(n-2)

数列{an}的通项公式为an=3×2^(n-2).

2.

bn=(2/3)nan=(2/3)×n×3×2^(n-2)=n×2^(n-1)

Sn=b1+b2+...+bn=1×2^0+2×2^1+...+n×2^(n-1)

2Sn=1×2^1+2×2^2+...+(n-1)×2^(n-1)+n×2ⁿ

Sn-2Sn=-Sn=2^0+2^1+...+2^(n-1)-n×2ⁿ=(2ⁿ-1)/(2-1) -n×2ⁿ=(1-n)×2ⁿ -1

Sn=(n-1)×2ⁿ +1