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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
avatar+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)))))))))

Jeffes02  Aug 9, 2017
 #2
avatar+90023 
+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"...!!!

 

 

cool cool cool

CPhill  Aug 9, 2017
edited by CPhill  Aug 9, 2017

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