等差数列|an|的各项均为正数,a1=3,前n项和为Sn,|bn|为等比数列,b1=1,且b2S2=64,b3S3=960.(1)求an与bn;（2）另cn=an*bn,求{cn}的前n项和Tn

Question

等差数列|an|的各项均为正数,a1=3,前n项和为Sn,|bn|为等比数列,b1=1,且b2S2=64,b3S3=960.
(1)求an与bn;
（2）另cn=an*bn,求{cn}的前n项和Tn

Answer 1

设公差为d,公比为q
q*(3+3+d)=64
q*q*(3+3+d+3+d+d)=960
q=64/(6+d)
4096*(9+3d)=960*(d*d+12d+36)
64*(9+3d)=15*(d*d+12d+36)
64*(3+d)=5*(d*d+12d+36)
192+64d=5d*d+60d+180
5d*d-4d-12=0
d=2或者d=-6/5舍去
q=8
an=2n+1
bn=8^(n-1)
cn=(2n+1)*8^(n-1)
Tn=3*8^0+5*8^1+...+(2n+1)*8^(n-1)
8Tn= 3*8^1+5*8^2+...+(2n+1)*8^n
7Tn=(2n+1)*8^n-{2*8^1+2*8^2+..+2*8^(n-1)}-3*8^0
Tn=1/7*{(2*8^n-37)/7+(2n+1)*8^n}