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已知数列{an},{bn}满足设a1=2,a(n+1)=2/(an+1),bn=|(an+2)/(an-1).求使|a(n-1)|
题目详情
已知数列{an},{bn}满足设a1=2,a(n+1)=2/(an+1),bn=|(an+2)/(an-1).求使|a(n-1)|
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答案和解析
b1=(a1+2)/(a1-1)=4
代入可得bn=(-1/2)b(n-1)
bn=b1(-1/2)^(n-1)=(-1/2)^(n-3)
(an+2)/(an-1)=(-1/2)^(n-3)
an+2=(an-1)(-1/2)^(n-3) =an(-1/2)^(n-3) -(-1/2)^(n-3)
an[1-(-1/2)^(n-3)]=-2-(-1/2)^(n-3)
an=-[2+(-1/2)^(n-3)]/[1-(-1/2)^(n-3)]
代入可得bn=(-1/2)b(n-1)
bn=b1(-1/2)^(n-1)=(-1/2)^(n-3)
(an+2)/(an-1)=(-1/2)^(n-3)
an+2=(an-1)(-1/2)^(n-3) =an(-1/2)^(n-3) -(-1/2)^(n-3)
an[1-(-1/2)^(n-3)]=-2-(-1/2)^(n-3)
an=-[2+(-1/2)^(n-3)]/[1-(-1/2)^(n-3)]
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