Question

数列{an}满足1/2a1+1/2^2a2+1/2^3a3……1/2^nan=2n+5. 要过程!

Answer 1

(1)

1/2a1+1/2^2a2+1/2^3a3+……+1/2^nan=2n+5 ①

当n=1时,1/2a1=2+5=7,

∴ a1=14

(2)

当n≥2时,

1/2a1+1/2^2a2+1/2^3a3+……+1/2^(n-1)a(n-1)=2(n-1)+5 ②

①-②:

1/2ⁿ*an=2,

∴an=2^(n+1)

∴{an}通项公式

an={14,n=1

{ 2^(n+1) ,(n≥2)

(3)

S1=a1=14

当n≥2时,

Sn=a1+a2+.+an

=14+[8+16+.+2^(n+1)]

=14+8[2^(n-1)-1]

=6+2^(n+2)

n=1时上式也成立

∴Sn=6+2^(n+2)