<s>[INST] Find the nth term of a recursive sequence I have a the following sequence:
$$\begin{gather}
a_1 = 3 \\
a_{n + 1} = 1 + \frac{a_n}{2}
\end{gather}
$$
How can I find the $a_n$ term? 
 [/INST] Put $n = 1$; you get $a_2 = 1 + \frac{1}{2} * 3 = \frac{5}{2}$
Following this continually, you get:
$$a_{n} = \frac{3 + \Sigma {2^{n - 1}}}{2^{n - 1}}$$
where, $n = 1, 2, 3 .. $
Aletnatively
You have $a_2 = 1 + \frac{a_1}{2}$
$$a_3 = 1 + \frac{a_2}{2} = 1 + \frac{1}{2} * (1 + \frac{a_1}{2}) = 1 + \frac{1}{2} + \frac{a_1}{4}$$
Hence, in this manner, you generate:
$$a_{n + 1} = 1 + \frac{1}{2} + ... \text{n terms} + \frac{a_1}{2^n} = \frac{1 * (1 - \frac{1}{2^n}) }{1 - \frac{1}{2}} + \frac{3}{2^n}$$
Therefore, you conclude:
$$a_{n + 1} = 2 + \frac{1}{2^n}$$
where $n \ge 1$
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