Question

等差内容

等差数列an bn cn dn 前n项和记为Sn Tn Pn Qn，Sn/Tn=5n+1/3n-1. f(n)=an/bn. cn/dn=5n-2/3n-2 g(n)=Pn/Qn f(n)/g(n)的最小值为多少

Answer 1

解：当n=1时 S1/T1=a1/b1=3 ①
当n=2时 S2/T2=(a1+a2)/(b1+b2)=11/5 ②
当n=3时 S3/T3=（a1+a3）/(b1+b3)=2 ③
联立①②③得 a1=d2 b1=1/3d2 d1=5/3d2
f(n)=an/bn=(a1+(n-1)d1)/(b1+(n-1)d2)=5n-2/3n-2
因为an/bn=cn/dn
所以Sn/Tn=Pn/Qn
g(n)=5n+1/3n-1
f(n)/g(n)=[(5n-2)(3n-1)]/[(3n-2)(5n+1)]
=【15n²-11n+2】/【15n²-7n-2】
=1-4(n-1)/[(3n-2)(5n+1)]
令m(x)=(n-1)/[15n²-7n-2]
m'(x)={[15n²-7n-2]-(n-1)(30n-7)}/[15n²-7n-2]²
=【15n²-7n-2-30n²+37n-7】/[15n²-7n-2]²
=【-15n²+30n-9】/[15n²-7n-2]²

Answer 2