Question

在数列｛an｝中,a1=2/3,an+1=2an/an+1 证明｛1/an -1｝为等比数列 求数列｛（2n-1）/an

Answer 1

(1)

a(n+1)=2an/(an +1)

1/a(n+1)=(an+1)/(2an)=(1/2)(1/an)+1/2

1/a(n+1) -1=(1/2)(1/an)-1/2=(1/2)(1/an -1)

[1/a(n+1)-1]/(1/an -1)=1/2,为定值.

1/a1-1=1/(2/3)-1=1/2

数列{1/an -1}是以1/2为首项,1/2为公比的等比数列.

(2)

1/an -1=1/2ⁿ

1/an=1+1/2ⁿ

(2n-1)/an=(2n-1)(1+1/2ⁿ)=(2n-1)+(2n-1)/2ⁿ=2n-1-1/2ⁿ+n/2^(n-1)

Sn=2(1+2+...+n)-n-(1/2)(1-1/2ⁿ)/(1-1/2)+1/1+2/2+3/2²+...+n/2^(n-1)

=n²+1/2ⁿ-1+[1+2/2+3/2²+...+n/2^(n-1)]

令Cn=1+2/2+3/2²+...+n/2^(n-1)

则Cn/2=1/2+2/2²+...+(n-1)/2^(n-1)+n/2ⁿ

Cn-Cn/2=Cn/2=1+1/2+1/2²+...+1/2^(n-1)-n/2ⁿ

=(1-1/2ⁿ)/(1-1/2)-n/2ⁿ

=2(1-1/2ⁿ)-n/2ⁿ

Cn=4-4/2ⁿ-2n/2ⁿ

Sn=n²+1/2ⁿ-1+4-4/2ⁿ-2n/2ⁿ=n²-(2n+3)/2ⁿ +3