Question

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|〈

Answer 1

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)