Given the sequence an = [1 / a (n-1)] + 2, A1 = 2, find the general term formula of the sequence

Given the sequence an = [1 / a (n-1)] + 2, A1 = 2, find the general term formula of the sequence

When n ≥ 2,
an=1/a(n-1) +2=[2a(n-1)+1]/a(n-1)
an +√2-1=[(√2+1)a(n-1)+1]/a(n-1)=(√2+1)[a(n-1)+(√2-1)]/a(n-1)
an-√2-1=[(1-√2)a(n-1)+1]/a(n-1)=-(√2-1)[a(n-1)-√2-1]/a(n-1)
(an+√2-1)/(an-√2-1)=-(3+2√2)[a(n-1)+(√2-1)]/[a(n-1)-√2-1]
[(an + √ 2-1) / (an - √ 2-1)] / {[[a (n-1) + (√ 2-1)] / [[a (n-1) - √ 2-1]} = - (3 + 2 √ 2), is a constant value
(a1+√2-1)/(a1-√2-1)=(2+√2-1)/(2-√2-1)=(√2+1)/(1-√2)=-(3+2√2)
The sequence {(an + √ 2-1) / (an - √ 2-1)} is an equal ratio sequence with - (3 + 2 √ 2) as the first term and (3 + 2 √ 2) as the common ratio
(an+√2-1)/(an-√2-1)=[-(3+2√2)]ⁿ
[-(3+2√2)]ⁿan -(√2+1)[-(3+2√2)]ⁿ=an+√2-1
{1-[-(3+2√2)]ⁿ}an=1-√2-(-1)ⁿ·(√2+1)^(2n+1)
an=[1-√2-(-1)ⁿ·(√2+1)^(2n+1)]/{1-[-(3+2√2)]ⁿ}