若无穷数列an满足a1=2,an+1=根号3an+2,且有极限,则liman=
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![若无穷数列an满足a1=2,an+1=根号3an+2,且有极限,则liman=](/uploads/image/z/14293311-15-1.jpg?t=%E8%8B%A5%E6%97%A0%E7%A9%B7%E6%95%B0%E5%88%97an%E6%BB%A1%E8%B6%B3a1%3D2%2Can%2B1%3D%E6%A0%B9%E5%8F%B73an%2B2%2C%E4%B8%94%E6%9C%89%E6%9E%81%E9%99%90%2C%E5%88%99liman%3D)
若无穷数列an满足a1=2,an+1=根号3an+2,且有极限,则liman=
若无穷数列an满足a1=2,an+1=根号3an+2,且有极限,则liman=
若无穷数列an满足a1=2,an+1=根号3an+2,且有极限,则liman=
A(n+1) = √3*An + 2
A(n+1) + c = √3*An + 2 + c = √3 * [An + (2+c)/√3 ]
令 c = (2+c)/√3,则 c = 2/(√3 -1) = √3 + 1
所以有:
A(n+1) + √3 + 1 = √3 (An + √3 + 1)
可见,{An+√3+1}就是一个等比数列,因此有:
An+√3+1 = (A1 + √3 + 1) * (√3)^(n-1) = (√3+2) * (√3)^(n-1)
所以,An = (√3+2)*(√3)^(n-1) - √3 - 1
limAn 为无穷大.
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