Question

已知数列{a n }的前n项和为S n ,且S n =2n 2 +n,n∈N * ,数列{b n }满足a n =4log 2 b n +3,n∈N * .(1)

已知数列{a n }的前n项和为S n ,且S n =2n 2 +n,n∈N * ,数列{b n }满足a n =4log 2 b n +3,n∈N * .(1)求a n ,b n ;(2)求数列{a n ·b n }的前n项和T n .

Answer 1

(1) a n =4n-1,n∈N *   b n =2 n-1 ,n∈N *   (2) T n =(4n-5)2 n +5,n∈N *

解:(1)由S n =2n 2 +n,得 当n=1时,a 1 =S 1 =3; 当n≥2时,a n =S n -S n-1 =4n-1 所以a n =4n-1,n∈N * 由4n-1=a n ="4" log 2 b n +3,得b n =2 n-1 ,n∈N * . (2)由(1)知a n b n =(4n-1)·2 n-1 ,n∈N * , 所以T n =3+7×2+11×2 2 +…+(4n-1)·2 n-1 , 2T n =3×2+7×2 2 +…+(4n-5)·2 n-1 +(4n-1)·2 n , 所以2T n -T n =(4n-1)2 n -[3+4(2+2 2 +…+2 n-1 )]=(4n-5)2 n +5 故T n =(4n-5)2 n +5,n∈N * .