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What is \(1 + \cfrac{3}{1 + \cfrac{3}{1 + \cfrac{3}{1 + \dotsb}}}\) equal to?

 Dec 10, 2020
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Solve for x:
x = 1 + 3/x

Bring 1 + 3/x together using the common denominator x:
x = (x + 3)/x

Cross multiply:
x^2 = x + 3

Subtract x + 3 from both sides:
x^2 - x - 3 = 0

Add 3 to both sides:
x^2 - x = 3

Add 1/4 to both sides:
x^2 - x + 1/4 = 13/4

Write the left hand side as a square:
(x - 1/2)^2 = 13/4

Take the square root of both sides:
x - 1/2 = sqrt(13)/2 or x - 1/2 = -sqrt(13)/2

Add 1/2 to both sides:
x = 1/2 + sqrt(13)/2 or x - 1/2 = -sqrt(13)/2

Add 1/2 to both sides:
 
x = 1/2 + sqrt(13)/2 ==2.3027756377.........etc.

 Dec 11, 2020

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