已知An的前n项和为Sn且Sn=2n²+n,bn满足An=4log2bn+3.求⑴An已知An的前n项和为Sn且Sn=2n²+n,bn满足An=4log2bn+3.求⑴Anbn⑵求An×bn的前n项和Tn

Question

已知An的前n项和为Sn且Sn=2n²+n,bn满足An=4log2bn+3.求⑴An
已知An的前n项和为Sn且Sn=2n²+n,bn满足An=4log2bn+3.
求⑴An bn
⑵求An×bn的前n项和Tn

Answer 1

Sn=2n^2+n
n=1,a1=3
for n>=2
an = Sn - S(n-1)
=2(2n-1) +1
=4n -1
an = 4logbn +3
logbn = n- 1
bn = 2^(n-1)
let
S =1.2^1+2.2^2+...+n.2^n (1)
2S = 1.2^2+2.2^3+...+n.2^(n+1) (2)
(2)-(1)
S = n.2^(n+1)-(2+2^2+...+2^n)
=n.2^(n+1)-2(2^n -1)
cn =an.bn
=(4n-1).2^(n-1)
= 2(n.2^n) - 2^(n-1)
Tn =c1+c2+...+cn
=2S - (2^n -1)
=2n.2^(n+1)-5(2^n -1)
= 5 + (4n-5).2^n