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
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![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](/uploads/image/z/11519726-14-6.jpg?t=%E8%AE%BE%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%85%AC%E6%AF%94%E4%B8%BAq%2C%E5%89%8D%E9%A1%B9%E5%92%8C%E4%B8%BAsn%2C%E6%B1%82%E6%95%B0%E5%88%97%7Bsn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8Cun)
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)]