Determine the value of the periodic continued fraction x=0.5+0.10.5+0.10.5+…x = 0.5 + \frac{0.1}{0.5 + \frac{0.1}{0.5 + \dots}}x=0.5+0.5+0.5+…0.10.1.
x=1+62x = \frac{1 + \sqrt{6}}{2}x=21+6
x=1+34x = \frac{1 + \sqrt{3}}{4}x=41+3
x=1+22x = \frac{1 + \sqrt{2}}{2}x=21+2
x=1+54x = \frac{1 + \sqrt{5}}{4}x=41+5