设首项为a,公差为d的等差数列前n项和为A n,又首项为a,公比为r的等比数列的前n项和为Gn,其中a不等于0,|r|〈1,令 Sn=G1+G2+.+Gn,若有lim(n趋向于无穷大)(An/n-Sn)=a 求r值.
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![设首项为a,公差为d的等差数列前n项和为A n,又首项为a,公比为r的等比数列的前n项和为Gn,其中a不等于0,|r|〈1,令 Sn=G1+G2+.+Gn,若有lim(n趋向于无穷大)(An/n-Sn)=a 求r值.](/uploads/image/z/12514276-28-6.jpg?t=%E8%AE%BE%E9%A6%96%E9%A1%B9%E4%B8%BAa%2C%E5%85%AC%E5%B7%AE%E4%B8%BAd%E7%9A%84%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BAA+n%2C%E5%8F%88%E9%A6%96%E9%A1%B9%E4%B8%BAa%2C%E5%85%AC%E6%AF%94%E4%B8%BAr%E7%9A%84%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BAGn%2C%E5%85%B6%E4%B8%ADa%E4%B8%8D%E7%AD%89%E4%BA%8E0%2C%7Cr%7C%E3%80%881%2C%E4%BB%A4+Sn%3DG1%2BG2%2B.%2BGn%2C%E8%8B%A5%E6%9C%89lim%28n%E8%B6%8B%E5%90%91%E4%BA%8E%E6%97%A0%E7%A9%B7%E5%A4%A7%29%28An%2Fn-Sn%29%3Da+%E6%B1%82r%E5%80%BC.)
设首项为a,公差为d的等差数列前n项和为A n,又首项为a,公比为r的等比数列的前n项和为Gn,其中a不等于0,|r|〈1,令 Sn=G1+G2+.+Gn,若有lim(n趋向于无穷大)(An/n-Sn)=a 求r值.
设首项为a,公差为d的等差数列前n项和为A n,又首项为a,公比为r的等比数列的前n项和为Gn,其中a不等于0,|r|〈1,令 Sn=G1+G2+.+Gn,若有lim(n趋向于无穷大)(An/n-Sn)=a 求r值.
设首项为a,公差为d的等差数列前n项和为A n,又首项为a,公比为r的等比数列的前n项和为Gn,其中a不等于0,|r|〈1,令 Sn=G1+G2+.+Gn,若有lim(n趋向于无穷大)(An/n-Sn)=a 求r值.
Gn=a(r^n-1)/(r-1)
Sn=(1/(r-1))(a(r+r^2+r^3+...+r^n)-(a+a+a+...+a))
=(1/(r-1))(a(r^n-1)/(r-1)-na)=(a/(r-1)^2)(r^n-1-n(r-1))
An=a+(1/2)n(n-1)d
(An/n)-Sn=(a/n)+(1/2)(n-1)d)-(a(r^n/(r-1)^2)+(1/(r-1)^2)+(a*n/(r-1))
n趋向于无穷大
=(1/2)nd-(d/2)+(1/(r-1)^2)+(a*n/(r-1))
=n*((d/2)+(a/(r-1))+(1/(r-1)^2)-(d/2)
=a
所以:
(d/2)+(a/(r-1))=0
(1/(r-1)^2)-(d/2)=a
a(1-r)^2+a(1-r)-1=0
1-r=((a^2+4a)^(1/2)-a)/(2a)
r=1-((a^2+4a)^(1/2)-a)/(2a)