What is the numerical value of x: $x = \frac{1}{3-\frac{1}{3-\frac{1}{3-\ldots}}}$
+128408 Note that we can write this as :
1
x = _______ multiply both sides by 3 - x
3 - x
x ( 3 - x) = 1 simplify
3x - x^2 = 1 rearrange as
x^2 - 3x + 1 = 0 complete the square on x
x^2 - 3x + 9/4 = -1 + 9/4
( x - 3/2)^2 = 5/4 take both roots
x -3/2 = ± sqrt (5) / 2
x = ±sqrt (5)/ 2 + 3/2
x = sqrt (3) /2 + 5/2 this value of x makes the continued fraction negative.....reject
x = -sqrt (5) / 2 + 3/2 = [3 - sqrt 5] / 2
