can anyone help me with this problem as well as give a clear explanation? thank you!!

Find the value of

\(x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}.\)

Guest Aug 9, 2017

#1**+1 **

I didn't manage to write the answer of this question by any mathematical ways, nor by using my hand, but the function gets closer and closer to \(≈ 1.41421356237\) as more layers are added, which equals to \(\sqrt2\).

This is also the reason that \(\sqrt2\) is an irrational number instead of a rational one, since it needs infinite fractions to express it.

Program: http://www.wolframalpha.com/input/?i=1%2B1%2F(2%2B1%2F(2%2B1%2F(2%2B1%2F(2%2B1%2F(2%2B1%2F(2%2B1%2F(2%2B1%2F(2%2B1%2F(2%2B1)))))))))

Jeffes02
Aug 9, 2017

#2**+1 **

Add 1 to both sides

x + 1 = 2 + [ 1 / [ 2 + [ 1 / [ 2 + 1 / [ ... ]

Now...let x + 1 = y

But...y = [ 2 + [ 1 / [ 2 + 1 / [ ... ]

So...we have that

y = 2 + [1 / y ] multiply through by y

y^2 = 2y + 1 rearrange

y^2 - 2y - 1 = 0

Solving this with the quadratic formula gives that y = 1 + √2 or y = 1 - √2

But since the right side of the original problem is positive then so is x

And since x + 1 = y then x = y - 1

So y must = 1 + √2

So...... x = [ 1 + √2 ] - 1 = √2

P.S. - thanks to one of our members - geno - for showing me this "trick"...!!!

CPhill
Aug 9, 2017