Given `a_1=1/2(a_0+A/(a_0)), a_2=1/2(a_1+A/(a_1))` and `a_(n+1)=1/2(a_n+A/(a_n))` for `n>=2`, where `a>0,A>0`. prove that `(a_n-sqrt(A))/(a_n+sqrt(A))=((a_1-sqrt(A))/(a_1+sqrt(A)))2^(n-1)`.

Explanation

Let :

P(n): frac{a_{n}-sqrt{A}}{a_{n}+sqrt{A}}=(frac{a_{1}-sqrt{A}}{a_{1}+sqrt{A}})^{20-1}

Step I

`P(1):(frac{a_{1}-sqrt{A}}{a_{1}+sqrt{A}})=(frac{a_{1}-sqrt{A}}{a_{1}+sqrt{A}})^{2-1}` (which is true)

`P(2):(frac{a_{2}-sqrt{A}}{a_{2}+sqrt{A}})=(frac{frac{1}{2}(a_{1}+frac{A}{a_{1}})-sqrt{A}}{frac{1}{2}(a_{1}+frac{A}{a_{1}})+sqrt{A}})=(frac{a_{1}+frac{A}{a_{1}}-2 sqrt{A}}{a_{1}+frac{A}{a_{1}}+2 sqrt{A}})=(frac{a_{1}+A-2 sqrt{A}}{a_{1}+A+2 sqrt{A}})=(frac{a_{1}-sqrt{A}}{a_{1}+sqrt{A}})^{23-1}`


Thus, `P(1)` and `P(2)` are true .

Step `I I`

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