等比数列{an}公比为q,前n项和为Sn,且S5,S10,S15成等差数列.1.求数列{nq^5n}前n项和Tn 2.证明:2S5,S10,S20-S10成等比数列
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![等比数列{an}公比为q,前n项和为Sn,且S5,S10,S15成等差数列.1.求数列{nq^5n}前n项和Tn 2.证明:2S5,S10,S20-S10成等比数列](/uploads/image/z/13295480-32-0.jpg?t=%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%7Ban%7D%E5%85%AC%E6%AF%94%E4%B8%BAq%2C%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E4%B8%94S5%2CS10%2CS15%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97.1.%E6%B1%82%E6%95%B0%E5%88%97%7Bnq%5E5n%7D%E5%89%8Dn%E9%A1%B9%E5%92%8CTn+2.%E8%AF%81%E6%98%8E%EF%BC%9A2S5%2CS10%2CS20-S10%E6%88%90%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97)
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等比数列{an}公比为q,前n项和为Sn,且S5,S10,S15成等差数列.1.求数列{nq^5n}前n项和Tn 2.证明:2S5,S10,S20-S10成等比数列
等比数列{an}公比为q,前n项和为Sn,且S5,S10,S15成等差数列.
1.求数列{nq^5n}前n项和Tn
2.证明:2S5,S10,S20-S10成等比数列
等比数列{an}公比为q,前n项和为Sn,且S5,S10,S15成等差数列.1.求数列{nq^5n}前n项和Tn 2.证明:2S5,S10,S20-S10成等比数列
1.
q=1时,S5=5 S10=10 S15=15
2S10=20 S5+S15=5+15=20 2S10=S5+S15,S5、S10、S15成等差数列,满足题意.
q≠1时,
2S10=S5+S15
2a1(q^10-1)/(q-1)=a1(q^5-1)/(q-1)+a1(q^15-1)/(q-1)
整理,得
2q^10=q^5+q^15
(q^5)^3-2(q^5)^2+q^5=0
q^5(q^5 -1)^2=0
q^5=0(舍去)或q^5=1(舍去)
综上,得q=1
nq^5n=n
Tn=1+2+...+n=n(n+1)/2
2.
2S5=2×5=10 S10=10 S20-S10=10
S10/(2S5)=(S20-S10)/S10=1,为定值.
2S5、S10、S20-S10是公比为1的等比数列.