Question

Evaluate the given fraction 

7176f36c12fa484384fa7989094be80d.docx

Answer 1

$Let\ x = 3+ \frac {1}{2+\frac{1}{3+\frac{1}{2+\frac{1}{3+....}}}} \\ \\ x = 3+ \frac {1}{2+\frac{1}{(3+\frac{1}{2+\frac{1}{3+....}})}} \\ \\ x = 3+ \frac {1}{2+\frac{1}x} \\ \\ x = 3+ \frac {1}{\frac{2x+1}x} \\ \\ x = 3+ \frac {x}{2x+1} \\ \\ x = \frac {3(2x+1)+x}{2x+1} = \frac{7x+3}{2x+1} \\ \\ x(2x+1) = 7x+3 \\ \\ 2x^2+x - 7x -3 =0 \\ 2x^2 -6x -3 = 0 \\ \\ \Delta = 6^2+4*2*3 = 60 \\ \\ \sqrt{Delta} = 2\sqrt{15} \\ \\ x = \frac{(6 + 2\sqrt{15} )}{2*2} = \frac{(3+ \sqrt{15} )}{2}$

we ignore the negative root of quadratic equation in x, as the given x is positive.