Sn是等比数列的前n项和,求Sn的前n项和
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Sn是等比数列的前n项和,求Sn的前n项和
![Sn是等比数列的前n项和,求Sn的前n项和](/uploads/image/z/16553895-15-5.jpg?t=Sn%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%2C%E6%B1%82Sn%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C)
设等比数列首项为a1,公比为q,则a1≠0,q≠0.
q=1时,Sn=na1
Tn=S1+S2+...+Sn=(1+2+...+n)a1=n(n+1)a1/2
q≠1时,Sn=a1(qⁿ-1)/(q-1)
Tn=S1+S2+...+Sn=[a1/(q-1)][(q-1)+(q²-1)+...+(qⁿ-1)]
=[a1/(q-1)][(q+q²+...+qⁿ)-n]
=[a1/(q-1)][q(qⁿ-1)/(q-1) -n]
=a1q(qⁿ-1)/(q-1)² -na1/(q-1)
q=1时,Sn=na1
Tn=S1+S2+...+Sn=(1+2+...+n)a1=n(n+1)a1/2
q≠1时,Sn=a1(qⁿ-1)/(q-1)
Tn=S1+S2+...+Sn=[a1/(q-1)][(q-1)+(q²-1)+...+(qⁿ-1)]
=[a1/(q-1)][(q+q²+...+qⁿ)-n]
=[a1/(q-1)][q(qⁿ-1)/(q-1) -n]
=a1q(qⁿ-1)/(q-1)² -na1/(q-1)
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