已知数列an是各项均不为0的等差数列,Sn为其前n项和,且满足S2n-1=1/2an^2,数列bn满足,当n为奇数时bn=2^
问题描述：
已知数列an是各项均不为0的等差数列,Sn为其前n项和,且满足S2n-1=1/2an^2,数列bn满足,当n为奇数时bn=2^
当n为奇数时bn=2^(n-1),当n为偶数时bn=1/2an-1(n-1在下边),Tn为bn的前n项和.(1)求an和bn.(2)试比较T2n与2n^2+n/3的大小
答
(1)、S2n-1=1/2an^2和an是各项均不为0的等差数列得
S1 =1/2 a1^2 =a1
a1=2
S3 =1/2 a2^2 =3a2
a2 = 6
所以 an = 4n-2
n为偶数时bn=1/2 an-1 =2n-3
(2)、T2n = b1+b2+b3 +……+ b(2n-1) + b2n
=(b1+b3++……+ b(2n-1)) + (b2+b4++……+ b(2n))
=(1- 4^n)/(-3) + (1+ 4n-3 )n/2
= (4^n -1)/3 + (2n-1)n
T2n—(2n^2+n/3) = (4^n -1 -4n )/3 >0 (n>1时)
T2n > 2n^2+n/3 （n>1）
T2n