数列an满足a1=1,n乘以an+1=(n+1),an+n乘以(n+1)(1)证明数列an/n是等差数(2)设bn=3的n次方乘以根号an求数列bn的欠n项和

Question

数列an满足a1=1,n乘以an+1=(n+1),an+n乘以(n+1)
(1)证明数列an/n是等差数 (2)设bn=3的n次方乘以根号an 求数列bn的欠n项和

Answer 1

na(n+1)=(n+1)an+n(n+1)
两边同除n(n+1)
a(n+1)/(n+1) = an/n + 1
则a(n+1)/(n+1)-an/n=1
所以an/n是等差数列
a1/1=1
an/n=1+(n-1)*1=n
an=n^2
bn=3^n*n
b1 = 3*1
b2=3^2*2
Sn=b1+b2+...+bn
=3*1+3^2*2+3^3*3+3^n*n (1)
3Sn = 0+3^2*1+3^3*2+...+3^n*(n-1)+3^(n+1)*n (2)
(2)-(1)
2Sn = -3*1-3^2-3^3-3^n+n*3^(n+1)
=-(3+3^2+...+3^n)+n*3^(n+1)
=-(3-3^(n+1))/(1-3)+n*3^(n+1)
=3/2-3^(n+1)/2+n*3^(n+1)
Sn = 3/4-(n/2-1/4)*3^(n+1)