在数列An中 A1=3 2An+1=(1+1/n)²An+2(n-1/n)1求an的通项公式 2令bn=An+1-An/2 求bn的前n项和

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在数列An中 A1=3 2An+1=(1+1/n)²An+2(n-1/n)1求an的通项公式 2令bn=An+1-An/2 求bn的前n项和
在数列An中 A1=3 2An+1=(1+1/n)²An+2(n-1/n)
1求an的通项公式  2令bn=An+1-An/2 求bn的前n项和

在数列An中 A1=3 2An+1=(1+1/n)²An+2(n-1/n)1求an的通项公式 2令bn=An+1-An/2 求bn的前n项和
(1)
a1=3
2a(n+1)=(1+1/n)^2.an+2(n-1/n)
2a(n+1)= [(n+1)^2/n^2]an + 2(n^2-1)/n
2a(n+1)/(n+1)^2 = an/n^2 + 2(n-1)/[n(n+1)]
a(n+1)/(n+1)^2 = an/(2n^2) + (n-1)/[n(n+1)]
=an/(2n^2) - 1/n + 2/(n+1)
a(n+1)/(n+1)^2 -2/(n+1) = (1/2)[ an/n^2 - 2/n ]
=> {an/n^2 - 2/n} 是等比数列,q=1/2
an/n^2 - 2/n = (1/2)^(n-1) .(a1 - 2)
=(1/2)^(n-1)
an = n^2[ 2/n + (1/2)^(n-1) ]
(2)
let
S= 1.(1/2)^1+2.(1/2)^2+...+n.(1/2)^n (1)
(1/2)S= 1.(1/2)^2+2.(1/2)^3+...+n.(1/2)^(n+1) (2)
(1)-(2)
(1/2)S = (1/2+1/2^2+...+1/2^n) -n.(1/2)^(n+1)
= (1- (1/2)^n ) -n.(1/2)^(n+1)
S = 2(1- (1/2)^n ) -n.(1/2)^n
bn= a(n+1) -an/2
= (n+1)^2[ 2/(n+1) + (1/2)^n ] - (n^2/2)[ 2/n + (1/2)^(n-1) ]
= 2(n+1)-n + (1/2)^n .[(n+1)^2 - n^2]
= n+2 + (2n+1)(1/2)^n
= (n+2) + (1/2)^n + 2(n.(1/2)^n)
Sn =b1+b2+...+bn
= (n+5)n/2 + (1-(1/2)^n ) + 2S
=(n+5)n/2 + 5(1-(1/2)^n ) -2n.(1/2)^n
=(n^2+5n+10)/2 - (2n+5).(1/2)^n