{an},{bn}是等差数列Sn,Tn分别为它们前n项的和Sn/Tn=(2n-3)/(4n-3)求a4/b7

Question

{an},{bn}是等差数列Sn,Tn分别为它们前n项的和Sn/Tn=(2n-3)/(4n-3)求a4/b7

Answer 1

a(n)=a+(n-1)c
b(n)=b+(n-1)d
s(n)=na+n(n-1)c/2,
t(n)=nb+n(n-1)d/2,
(2n-3)/(4n-3)=s(n)/t(n)=[na+n(n-1)c/2]/[nb+n(n-1)d/2]=[a+(n-1)c/2]/[b+(n-1)d/2],
n=1时,
(2-3)/(4-3)=-1=[a]/[b],b=-a.
(2n-3)/(4n-3)=[a+(n-1)c/2]/[-a+(n-1)d/2],
(2n-3)[-a+(n-1)d/2] = (4n-3)[a+(n-1)c/2],
-2na+3a+n(n-1)d-3(n-1)d/2 = 4na-3a+2n(n-1)c - 3(n-1)c/2,
n(n-1)[2c-d] + n[4a-3c/2 + 2a + 3d/2] + [-3a+3c/2 - 3a - 3d/2] = 0,
d = 2c,
0=4a-3c/2+2a+3d/2=6a-3c/2+3c=6a+3c/2,c=-2a,d=-4a.
a(n)=a-2(n-1)a,
b(n)=-a-4(n-1)a,
a(4)/b(7)=[a-2*3*a]/[-a-4*6*a]=[-5]/[-25] = 1/5