Question

在数列｛an｝中，a1等于1，an＋1等于（1＋1/n）an＋（n＋1）/2^n

Answer 1

1.
A(n+1)=(n+1)An/n+(n+1)/2^n
A(n+1)/(n+1)=An/n+1/2^n
依此类推
An/n=A(n-1)/(n-1)+1/2^(n-1)
A(n-1)/(n-1)=A(n-2)/(n-2)+1/2^(n-2)
……
A2/2=A1/1+1/2^1
上式相加，相同项消去
An/n=A1/1+(1/2^1+1/2^2+……+1/2^(n-1))
=1+1/2×(1-(1/2)^(n-1))/(1-1/2)
=2-1/2^n
Bn=2-1/2^n
2.
An/n=2-1/2^n
An=2n-n/2^n
{An}分为两部分，2n是等差数列，n/2^n是等差数列与等比数列相乘
第一部分求和
Sn1=(2+2n)n/2=(n+1)n
第二部分求和
Sn2=1/2+2/4+3/8+……+n/2^n
两边同乘2
2Sn2=1+2/2+3/4+……+n/2^(n-1)
两式错位相减
2Sn2-Sn2=1+[1/2+1/4+1/8……+1/2^(n-1)]-n/2^n
=1+1/2×(1-(1/2)^(n-1))/(1-1/2)-n/2^n
=2-(n+2)/2^n
Sn=Sn1-Sn2
=n^2+n-2+(n+2)/2^n
=(n-1)(n+2)+(n+2)/2^n