Question

一道数列的题,算了好久了a（下标1）=a,a（下标n+1）=3（a（下标n）+1）/（a（下标n）+3）,求a（下标n）

Answer 1

a1=a

a(n+1)=3(an +1)/(an +3)

a(n+1) + √3 = 3(an +1)/(an +3) + √3

= [(3+√3)an + (3√3+3) ]/(an +3)

=(3+√3)( an +√3) /(an +3)

1/[a(n+1) + √3] = (an + 3)/[(3+√3)( an +√3) ]

= 1/(3+√3) + (3-√3)/(an +√3)

1/[a(n+1) + √3] + 1/(3-√3) = (3-√3) [ 1/(an +√3) + 1/(3-√3)]

=>{1/(an +√3) + 1/(3-√3)} 是等比数列,q=(3-√3)

1/(an +√3) + 1/(3-√3) = (3-√3)^(n-1) .[1/a1 + 1/(3-√3)]

= (3-√3)^(n-1) .[1/a + 1/(3-√3)]

1/(an +√3) = (3-√3)^(n-1)/a + (3-√3)^(n-2) - 1/(3-√3)

an = -√3 + 1/[(3-√3)^(n-1)/a + (3-√3)^(n-2) - 1/(3-√3)]