Find the general term of the sequence $a_n = (0, 3, 1/2 , 5/3 , 2/3 , 7/5 , 3/4 , ...)$. Then find the limit of this sequence. I have an exercise in which I am stuck. The exercise is 
Find the general term of the sequence 
$$a_n = (0, 3, \dfrac{1}{2}, \dfrac{5}{3}, \dfrac{2}{3}, \dfrac{7}{5}, \dfrac{3}{4}, ...)$$. 
Then find the limit of this sequence.
So far, I have tried like this(Idk if its right)
First thing, I have written sequence as like this
$$a_n = (0, 3, \dfrac{1}{2}, \dfrac{5}{3}, \dfrac{2}{3}, \dfrac{7}{5}, \dfrac{3}{4}, ...)$$
$$indexes \rightarrow 1, 2, 3,\hspace{0.5mm} 4,\hspace{1mm}5, 6, \hspace{0.5mm}7, 8, \hspace{2.2cm}$$
Now, I can get/ divide indexes in odd and even. 
For Odd indexes:
 $$a_n = (0, \dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, ...)$$
$$indexes \rightarrow \hspace{4mm}1,\hspace{2mm} 2, \hspace{1mm}3,\hspace{2.5mm} 4,..\hspace{2cm}$$
For Even indexes:
 $$a_n = (3, \dfrac{5}{3}, \dfrac{7}{5}, ...)$$
$$indexes \rightarrow \hspace{4mm}1,\hspace{2mm} 2, \hspace{1mm}3,..\hspace{2cm}$$
So Now, I know the sequence of odd and even indexes(I think):
$$Odd: \dfrac{n}{n+1}\ and\ even\ \dfrac{n+2}{n}\ \forall n : n=2(k+1)$$
I have arrived here. I don't know how to proceed. Thanks For all help.

$$a_{2n+1}=\frac {2n+3}{2n+1} $$
$$a_{2n}=\frac {n}{n+1} $$
$$\lim a_{2n}=1=\lim a_{2n+1} \implies $$
$$\lim a_n=1$$
