设等比数列{an}的公比为q,前项和为sn,求数列{sn}的前n项和un

设等比数列{an}的公比为q,前项和为sn,求数列{sn}的前n项和un
设等比数列{an}的公比为q,前项和为sn,求数列{sn}的前n项和un

设等比数列{an}的公比为q,前项和为sn,求数列{sn}的前n项和un
Sn=a1*(1-q^n)/(1-q),可得a1=(1-q)*Sn/(1-q^n)
且
Un=S1+S2+S3+……+Sn=[(1-q)+(1-q^2)+(1-q^3)+……+(1-q^n)]*a1/(1-q)
=(n-q-q^2-q^3-……-q^n)*a1/(1-q)=[n-q*(1-q^n)/(1-q)]*a1/(1-q)
=[n*(1-q)-q*(1-q^n)]*Sn/[(1-q^n)*(1-q)]