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I have looked everywhere for some good guides, so the question is:

You probably know the Fibonacci sequence (1,1,2,3,5,8…) but there is a lesser-known sequence called (Pythagoras stairs) generated by a similar recursion formula.

This is a sequence of pairs (Xn,Yn) usually arranged as follows:

1 2

2 3

5 7

12 17

And so on……. It begins with (x1,y1)=(1,1) and the recursion formula is

X{n+1} = X{n} + Y{n}

Y{n+1} = X{n}+ X{n+1}

Prove by induction that always

Y^2 = 2 X^2 ± 1

Pythagoras used this equation to generate rational approximations to (2)^1/2

Thanks in advance for usefully discuss to solve for this interesting question

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# The idea about Pythagoras stairs?

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