已知通项公式求Sn1.已知an=n*4的n次方求Sn2.已知an=3/n(n+3)求Sn3.已知an=n(n+2)求Sn

Question

已知通项公式求Sn
1.已知an=n*4的n次方 求Sn
2.已知an=3/n(n+3) 求Sn
3.已知an=n(n+2) 求Sn

Answer 1

1、Sn=1*4^1+2*4^2+3*4^3……+n*4^n ①
4Sn=1*4^2+2*4^3+3*4^4……+(n-1)*4^n+n*4^(n+1) ②
②-①得3Sn=-1*4^1-4^2-4^3-……-4^n+n*4^(n+1)=-4(1-4^n)/(1-4)=4(1-4^n)/3所以Sn=4(1-4^n)/9
2、an=3/n(n+3)=1/n-1/(n+3),Sn=(1-1/4)+(1/2-1/5)+(1/3-1/6)+(1/4-1/7)+(1/5-1/8)+(1/6-1/9)+……+[1/(n-3)-1/n]+[1/(n-2)-1/(n+1)]+[1/(n-1)-1/(n+2)]+[1/n-1/(n+3)]=1+1/2+1/3-1/(n+1)-1/(n+2)-1/(n+3)=11/6-1/(n+1)-1/(n+2)-1/(n+3)
3、公式：1^2+2^2+3^2+……+n^2=n(n-1)(2n-1)/6
所以Sn=n(n-1)(2n-1)/6+（1+n)n