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