数列{an},{bn}满足a1=k,a(n+1)=(2/3)an+n-4,bn=(-1)^n(an-3n+21) 其中k为实数,n属于N+证明数列{an}不是等比数列,若{bn}是等比数列,求k的范围
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![数列{an},{bn}满足a1=k,a(n+1)=(2/3)an+n-4,bn=(-1)^n(an-3n+21) 其中k为实数,n属于N+证明数列{an}不是等比数列,若{bn}是等比数列,求k的范围](/uploads/image/z/3977255-47-5.jpg?t=%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%2C%EF%BD%9Bbn%EF%BD%9D%E6%BB%A1%E8%B6%B3a1%3Dk%2Ca%EF%BC%88n%2B1%EF%BC%89%3D%EF%BC%882%2F3%EF%BC%89an%2Bn-4%2Cbn%3D%EF%BC%88-1%EF%BC%89%5En%28an-3n%2B21%29+%E5%85%B6%E4%B8%ADk%E4%B8%BA%E5%AE%9E%E6%95%B0%2Cn%E5%B1%9E%E4%BA%8EN%2B%E8%AF%81%E6%98%8E%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E4%B8%8D%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%2C%E8%8B%A5%EF%BD%9Bbn%EF%BD%9D%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%2C%E6%B1%82k%E7%9A%84%E8%8C%83%E5%9B%B4)
数列{an},{bn}满足a1=k,a(n+1)=(2/3)an+n-4,bn=(-1)^n(an-3n+21) 其中k为实数,n属于N+证明数列{an}不是等比数列,若{bn}是等比数列,求k的范围
数列{an},{bn}满足a1=k,a(n+1)=(2/3)an+n-4,bn=(-1)^n(an-3n+21) 其中k为实数,n属于N+
证明数列{an}不是等比数列,若{bn}是等比数列,求k的范围
数列{an},{bn}满足a1=k,a(n+1)=(2/3)an+n-4,bn=(-1)^n(an-3n+21) 其中k为实数,n属于N+证明数列{an}不是等比数列,若{bn}是等比数列,求k的范围
a(1)=k,
a(2)=(2/3)a(1)+1-4=2k/3-3=(2k-9)/3,
a(3)=(2/3)a(2)+2-4=(2/3)(2k-9)/3-2 = [4k-36]/9
若[a(2)]^2=a(1)a(3),则
(2k-9)^2/9=k(4k-36)/9,
(2k-9)^2=k(4k-36),
4k^2-36k+81=4k^2-36k,
81=0矛盾.
因此,[a(2)]^2不等于a(1)a(3),{a(n)}不是等比数列.
a(n+1)=(2/3)a(n)+n-4,
a(n+1)+x(n+1)+y=(2/3)a(n)+n-4+xn+x+y=(2/3)a(n)+(x+1)n+x+y-4
=(2/3)[a(n)+3(x+1)/2*n + 3(x+y-4)/2]
x=3(x+1)/2,x=-3,
y=3(x+y-4)/2,y=12-3x=21.
a(n+1)-3(n+1)+21=(2/3)a(n)+n-4-3(n+1)+21=(2/3)a(n)-2n+14=(2/3)[a(n)-3n+21]
{a(n)-3n+21}是首项为a(1)-3+21=k+18,公比为2/3的等比数列.
a(n)-3n+21=(k+18)(2/3)^(n-1)
b(n)=(-1)^n[a(n)-3n+21]=(-1)^n*(k+18)(2/3)^(n-1)=(-k-18)(-2/3)^(n-1)
-k-18不为0,也即k不等于-18时,
{b(n)}是首项为(-k-18),公比为(-2/3)的等比数列.
1、
a(n+1)/an = 2/3 +(n-4)/an ①
a(n+2)/a(n+1) = 2/3 + (n-3)/a(n+1)②
若①=②
则 (n-4)/an = (n-3)/a(n+1)
解得:an=3(n-3)(n-4)/(n-6);
a(n+1) = 3(n-2)(n-3)/(n-5) ≠(2/3)an+n-4
所以 {an}不是等比数列。
2、
k≠19