已知等差数列｛an｝和｛bn｝前n项和为An和Bn,且An/Bn为7n+45/n+3,则使得an/bn为整数的n有几个,

Question

已知等差数列｛an｝和｛bn｝前n项和为An和Bn,且An/Bn为7n+45/n+3,则使得an/bn为整数的n有几个,

Answer 1

1.An/Bn=(7n+45)/(n+3)=(7n+7*3+24)/(n+3)=7+24/(n+3)
An/Bn为整数,只需要24/(n+3)为整数,又n+3>3,则(n+3)=4,6,8,12或24
得n=1,3,5,9或21
2.an-2=2-4/a(n-1)=[2a(n-1)-4]/a(n-1)
1/(an-2)=a(n-1)/[2a(n-1)-4]
=[a(n-1)-2+2]/2[a(n-1)-2]
=1/2+1/[a(n-1)-2]
1/(an-2)-1/[a(n-1)-2]=1/2
所以bn=1/(an-2)是等差数列
公差d=1/2
所以1/(an-2)=1/(a1-2)+(n-1)*1/2=1/2+n/2-1/2=n/2
an-2=2/n
an=2+2/n