$\frac{1}{2},\frac{5}{3},\frac{11}{8},\frac{27}{19},...$. Find its 10th term. Given this sequence, find its 10th term and its exact limit.
$$\frac{1}{2},\frac{5}{3},\frac{11}{8},\frac{27}{19},...$$
I've been stuck in this question forever. I can't find any relation between them. The answer for the 10th term is $\frac{5333}{3771}$

$$\frac{1}{2},\frac{5}{3},\frac{11}{8},\frac{27}{19},...$$
$$\frac{1}{2},\frac{5}{(2+1)},\frac{11}{(2+1+5)},\frac{27}{(2+1+5+11)},...$$
$$\frac{1}{2},\frac{(2+3)}{3},\frac{(3+8)}{8},\frac{(8+19)}{19},...$$
In this way I would define the series as follows:
$$\text{ for }  a_1=\frac{x_1}{y_1} = \frac{1}{2}$$
$$a_n := \frac{x_n}{y_n} = \frac{y_{(n-1)} + y_n}{x_{(n-1)}+y_{(n-1)}} $$
so 
$$a_n := \frac{x_n}{y_n} = \frac{y_{(n-1)} + y_n}{x_{(n-1)}+y_{(n-1)}} = \frac{y_{(n-1)} + \left(x_{(n-1)}+y_{(n-1)}\right)}{x_{(n-1)}+y_{(n-1)}} = \frac{2y_{(n-1)} + x_{(n-1)}}{x_{(n-1)}+y_{(n-1)}} =$$

$$ a_n = \left(1 + \frac{y_{(n-1)}}{y_{(n-1)}+x_{(n-1)}}\right)$$

the series until the 10th term:
$$\frac{1}{2},\frac{5}{3},\frac{11}{8},\frac{27}{19},\frac{65}{46}, \frac{157}{111}, \frac{379}{368}, \frac{915}{647}, \frac{2209}{1562},  \frac{5333}{3771}$$
