作业帮数列an满足a1=-1,a(n+1)=2an+3n-4(1)求证a(n+1)-an+3是等比数列（2）求an(3)求|a1|+|a2|+.+|an|=?
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/11 02:34:02
数列an满足a1=-1,a(n+1)=2an+3n-4(1)求证a(n+1)-an+3是等比数列（2）求an(3)求|a1|+|a2|+.+|an|=?
数列an满足a1=-1,a(n+1)=2an+3n-4
(1)求证a(n+1)-an+3是等比数列
（2）求an
(3)求|a1|+|a2|+.+|an|=?
数列an满足a1=-1,a(n+1)=2an+3n-4(1)求证a(n+1)-an+3是等比数列（2）求an(3)求|a1|+|a2|+.+|an|=?
根据
a = 2an + 3n -4
得到
an = 2a + 3(n-1) - 4 = 2a + 3n - 7
根据
a = 2an + 3n -4
得到
a - an + 3 = an + 3n -1
以及
an - a + 3 = a + 3(n-1) - 1 = a + 3n -4
将 an = 2a + 3n - 7 代入到 a - an + 3 = an + 3n -1 等式右端
a - an + 3 = 2a + 3n - 7 + 3n -1 = 2a + 6n - 8
至此
a - an + 3 = 2a + 6n - 8 = 2(a + 3n -4)
an - a + 3 = a + 3n -4
两式做比值运算
[a - an + 3 ]/an - a + 3 = 2
因此 a(n+1)-an+3是等比数列 公比为 q = 2
----------------------------------
a1=-1,a(n+1)=2an+3n-4
a2 = 2a1 + 3 - 4 = -2 + 3 - 4 = -3
a2 - a1 + 3 = -3 + 1 + 3 = 1 (此为等比数列的首项)
根据 a(n+1)-an+3是等比数列
a - an + 3 = [a2 - a1 + 3]*q^(n-1) = 2^(n-1)
a2 - a1 + 3 = 2^0
a3 - a2 + 3 = 2^1
a4 - a3 + 3 = 2^2
……
an - a + 3 = 2^(n-2)
以上共 n-1 个等式相加
an - a1 + 3(n-1) = 2^0 + 2^1 + …… + 2^(n-2)
an + 1 + 3n - 3 = 2^(n-1) -1
an = 2^(n-1) -3n + 1
根据 an = 2^(n-1) - 3n + 1 ,则
n = 1 2 3 4 时 an < 0
n ≥ 5 时 an > 0
n ≤ 4 时
|a1|+|a2|+.+|an|
= -(a1 + a2 + …… + an)
= -[2^0 + 2^1 + …… +2^(n-1)] + 3(1 + 2 …… + n) - 1*n
= -2^n + 1 + 3n(1+n)/2 - n
= n(3n+1)/2 - 2^n + 1
n = 4 时
S4 = 4(3*4+1)/2 - 2^4 + 1 = 11
n ≥ 5 时
|a1|+|a2|+.+|an|
=S4 + a5 + a6 + …… + a7
= 11 + [2^4 + 2^5 + …… + 2^(n-1)] - 3 (5 + 6 + …… + n) + (n-4)
= 11 + 2^4 * [2^(n-4) -1] - 3(5 + n)(n-4)/2 + (n-4)
= 11 + 2^n - 16 - 3(n-4)(n+5)/2 + (n-4)
= 2^n - 5 - (n-4)(3n+13)/2
timu 题目有点难啊
而且计算量很大，最起码半小时才能打字完