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Challenge 345: Forever Fractions

(i) Find the value of the expression

1+12+13+141+\frac{1}{2+\frac{1}{3+\frac{1}{4}}}

(ii) Express 29/12 in the form

a+1b+1c+1da+\frac{1}{b+\frac{1}{c+\frac{1}{d}}}

where a, b, c and d are positive integers.

(iii) What value does the expression below approach as the calculation is made more and more accurate by including more of the parts represented by “…”?

2+12+12+12+...2+\frac{1}{2+\frac{1}{2+\frac{1}{2+...}}}