# The idea about Pythagoras stairs?

1. Nov 15, 2008

### memomath

Hello everybody

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

Last edited: Nov 15, 2008
2. Nov 15, 2008

### memomath

Hi again

Hello everybody

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

$$X\underline{n+1}$$=$$X\underline{n}$$+$$Y\underline{n}$$

$$Y\underline{n+1}$$= $$X\underline{n}$$+$$X\underline{n+1}$$

Prove by induction that always

$$Y\acute{2}$$=2$$X\acute{2}$$ $$\pm1$$

Pythagoras used this equation to generate rational approximations to $$\sqrt{2}$$

Moreover, if you would like please open the enclosed to find the corrections/clarifications of the question.

Thanks

Last edited: Nov 25, 2008
3. Nov 16, 2008

### memomath

No comments or solution until now!!!

4. Nov 16, 2008

### HallsofIvy

Staff Emeritus
A lot of people will not open a Word file because they are notorious for carrying viruses.

5. Nov 16, 2008

### memomath

Dear HallsofIvy

Thanks for your reply, in fact I already wrote the problem in webpage and at the same time wrote in the word file so there two options

6. Nov 17, 2008

### memomath

Hi again

Hello everybody

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

$$X\underline{n+1}$$=$$X\underline{n}$$+$$Y\underline{n}$$

$$Y\underline{n+1}$$= $$X\underline{n}$$+$$X\underline{n+1}$$

Prove by induction that always

$$Y\acute{2}$$=2$$X\acute{2}$$ $$\pm1$$

Pythagoras used this equation to generate rational approximations to $$\sqrt{2}$$

Moreover, if you would like please open the enclosed to find the question as pdf form.

Thanks

Last edited: Nov 25, 2008
7. Nov 17, 2008

### natives

Okay here is how you do it...The method of mathematical induction for proofs follows the induction theorem which requires that for any relation dependent on variable N ,then if N=1,N=2 are TRUE and assuming N= k is TRUE then N=k+1 should also be TRUE...So if you test the Pythagoras' stairs for N=1 and N=2 then you find that it is TRUE...then assume that N=k is true such that y^2(k+1)=2x^2(k+1)+- 1...so using the above you should be able to prove that y^2(k+2)=2x^2(k+2)+-1...you can do this by taking the left hand side..You should be knowing that y(k+2)=2x(k+1)+y(k+1)..square this to have y^2(k+2) =4x^2(k+1)+4x(k+1)y(k+1)+y^2(k+1)...but from previously assuming N=k is true we had y^2(k+1)=2x^2(k+1)+-1..substitute in the previous equation...also keep in mind x(k+2)=x(k+1)+y(k+1)...use this [square it] in the above equations to get the right hand side of y^2(k+2)=2x^2(k+2)+-1 then you have proven using the induction theorem.

Last edited: Nov 17, 2008
8. Nov 24, 2008

### memomath

some wrongs here in the question

YES I AGREE WITH YOU there are mistake in the question

Last edited by a moderator: Nov 26, 2008
9. Nov 24, 2008

### girlmatrix

Re: some wrongs here in the question

The above answer is not correct and there are some mistakes in the question, its easy to find the mistakes in the question.

However the hint for the right question should use these formulae

Y(n + 1) X(n + 1) = (2X(n+1) + Y(n+3) (X(n+4) + Y(n+5))

Last edited by a moderator: Nov 26, 2008