在数列｛an｝中,a₁=1,a(n+1）=[(n+1)/n]an+(n+1)/2ⁿ(1)设bn=an/n,求数列｛bn｝的通项公式（2）求数列｛an｝的前n项和Sn

在数列｛an｝中,a₁=1,a(n+1）=[(n+1)/n]an+(n+1)/2ⁿ(1)设bn=an/n,求数列｛bn｝的通项公式（2）求数列｛an｝的前n项和Sn
在数列｛an｝中,a₁=1,a(n+1）=[(n+1)/n]an+(n+1)/2ⁿ
(1)设bn=an/n,求数列｛bn｝的通项公式
（2）求数列｛an｝的前n项和Sn

在数列｛an｝中,a₁=1,a(n+1）=[(n+1)/n]an+(n+1)/2ⁿ(1)设bn=an/n,求数列｛bn｝的通项公式（2）求数列｛an｝的前n项和Sn
(1)
a(n+1)=[(n+1)/n]an+(n+1)/2ⁿ
a(n+1)/(n+1)=an/n+1/2ⁿ
a(n+1)/(n+1)+1/2^n=an/n+1/2^(n-1)
数列{an/n+1/2^(n-1)}各项均相等.
a1/1+1/2^(1-1)=1+1=2
an/n+1/2^(n-1)=2
an/n=2-2/2ⁿ
bn=2-2/2ⁿ
数列{bn}的通项公式为bn=2-2/2ⁿ
(2)
an=2n-2n/2ⁿ
令Tn=1/2^1+2/2^2+...+n/2^n
则Tn/2=1/2^2+2/2^3+...+(n-1)/2^n+n/2^(n+1)
Tn-Tn/2=Tn/2=1/2^1+1/2^2+...+1/2^n-n/2^(n+1)
=(1/2)(1-1/2ⁿ)/(1-1/2)-n/2^(n+1)
=1-1/2ⁿ-n/2^(n+1)
Tn=2-2/2ⁿ-n/2ⁿ
Sn=2(1+2+...+n)-2Tn
=n(n+1)-4+4/2ⁿ+2n/2ⁿ
=n²+n-4+(2n+4)/2ⁿ
=n²+n-4+(n+2)/2^(n-1)
^表示指数.