设数列an满足a1=4,an=(1/3)(an-1 +2),求lim(a2+a4+...+a2n -3n)我算出来an=3*(1/3)^n+3

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设数列an满足a1=4,an=(1/3)(an-1 +2),求lim(a2+a4+...+a2n -3n)我算出来an=3*(1/3)^n+3
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设数列an满足a1=4,an=(1/3)(an-1 +2),求lim(a2+a4+...+a2n -3n)我算出来an=3*(1/3)^n+3
设数列an满足a1=4,an=(1/3)(an-1 +2),求lim(a2+a4+...+a2n -3n)
我算出来an=3*(1/3)^n+3

设数列an满足a1=4,an=(1/3)(an-1 +2),求lim(a2+a4+...+a2n -3n)我算出来an=3*(1/3)^n+3
an=(1/3)[a(n-1)+2]
3an=a(n-1)+2
3an -3=a(n-1) -1
(an -1)/[a(n-1) -1]=1/3,为定值.
a1 -1=4-1=3,数列{an -1}是以3为首项,1/3为公比的等比数列.
an -1=3×(1/3)^(n-1)=1/3^(n-2)
an=1/3^(n-2) -1
考察数列偶数项第2k项:a(2k)=1/3^(2k -2) -1=(1/9)^(k-1) -1
a2+a4+...+a(2n)=1+(1/9)+...+(1/9)^(n-1) -n
=1×(1- 1/9ⁿ)/(1-1/9) -n
=(9/8)(1-1/9ⁿ)-n
a2+a4+...+a(2n)-3n=9/8 -1/[8×9^(n-1)] -4n
n->+∞,9/8为定值.1/[8×9^(n-1)]->0,-4n->-∞
lim[a2+a4+...+a(2n) -3n]=-∞
n->+∞

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