Question

已知an=2^n+n-2 求Sn 已知an=n*(1/3)^n 求Sn

Answer 1

an=2^n+n-2
Sn=2^1+1-2+2^2+2-2+.+2^n+n-2
=(2^1+2^2+2^3+.+2^n)+(1+2+3+.+n)-2*n
=2*(1-2^n)/(1-2)+n(n+1)/2-2n
=2*(2^n-1)+(n^2+n-4n)/2
=2^(n+1)-2+(n^2-3n)/2
=2^(n+1)+(n^2-3n)/2-2
an=n*(1/3)^n
Sn=1*(1/3)^1+2*(1/3)^2+3*(1/3)^3+.+n*(1/3)^n
1/3*Sn=1*(1/3)^2+2*(1/3)^3+3*(1/3)^4+.+n*(1/3)^(n+1)
Sn-1/3*Sn=(1/3)^1+(1/3)^2+(1/3)^3+.+(1/3)^n+n*(1/3)^(n+1)
2/3*Sn=[1-(1/3)^n]/2+n*(1/3)^(n+1)
2/3*Sn=1/2-(1/3)^n/2+n*(1/3)^(n+1)
2/3*Sn=1/2-(1/3)^n/2+n/3*(1/3)^n
2/3*Sn=1/2+n/3*(1/3)^n-(1/3)^n/2
2/3*Sn=1/2+(n/3-1/2)*(1/3)^n
Sn=3/4+3(n/6-1/4)*(1/3)^n
Sn=3/4+(n/6-1/4)*(1/3)^(n-1)