+0

0
333
2

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

Find the value of

$$x = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}.$$

Aug 9, 2017

#1
+178
+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)))))))))

Aug 9, 2017
#2
+94550
+1

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"...!!!

Aug 9, 2017
edited by CPhill  Aug 9, 2017