数列{An}中,A1>-1 ,且对任意的正整数n,An+1=An+2/An+1都成立.n属于正整数,比较An与根号2的大小.
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数列{An}中,A1>-1 ,且对任意的正整数n,An+1=An+2/An+1都成立.n属于正整数,比较An与根号2的大小.
数列{An}中,A1>-1 ,且对任意的正整数n,An+1=An+2/An+1都成立.n属于正整数,比较An与根号2的大小.
数列{An}中,A1>-1 ,且对任意的正整数n,An+1=An+2/An+1都成立.n属于正整数,比较An与根号2的大小.
a(n+1)-2^(1/2) = [a(n)+2]/[a(n)+1] - 2^(1/2) = {[1-2^(1/2)]a(n) + 2 - 2^(1/2)}/[a(n)+1]
=[1-2^(1/2)][a(n)-2^(1/2)]/[a(n)+1],
若a(1)=2^(1/2),则a(n)=2^(1/2).
若a(1)不等于2^(1/2),则a(n)不等于2^(1/2).[否则,若有a(k)=2^(1/2),则a(k-1)=2^(1/2),...,a(1)=2^(1/2).矛盾.]
此时,
1/[a(n+1)-2^(1/2)]=[1-2^(1/2)]^(-1)[a(n)+1]/[a(n)-2^(1/2)]
=[1-2^(1/2)]^(-1)[a(n)-2^(1/2)+1-2^(1/2)]/[a(n)-2^(1/2)]
=1/[a(n)-2^(1/2)] + 1/[1-2^(1/2)],
{1/[a(n)-2^(1/2)]}是首项为1/[a(1)-2^(1/2)],公差为1/[1-2^(1/2)]的等差数列.
1/[a(n)-2^(1/2)]= 1/[a(1)-2^(1/2)] + (n-1)/[1-2^(1/2)],
若-11 + [2^(1/2) - 1]/[a(1)-2^(1/2)]时,总有 n-1 > [2^(1/2)-1]/[a(1)-2^(1/2)],
(n-1)/[2^(1/2)-1] > 1/[a(1)-2^(1/2)],
0 > 1/[a(1)-2^(1/2)] + (n-1)/[1-2^(1/2)] = 1/[a(n)-2^(1/2)],
a(n) < 2^(1/2).
当n 2^(1/2)