Explicit formula for recurrence relation I am given recurrence relation : an = 5an/2 + 3n, for n = 2,4,8,16... and a2 = 1. 
I found first 4 terms and I don't see a pattern.
a4 = 5*1 + 3*4 = 17
a8 = 5*17 + 3*8 = 109
a16 = 5*109 + 3*16 = 593
a32 = 5*1 + 3*4 = 3061
Playing a bit with it on a paper and calculater I found out formula that seems to be correct is : 5n-1 - 3*2n + 2n+1, but this was just a lucky guess.. And of course I won't be able to use a calculator on upcoming final exam. 
Can anyone help me to solve such problems? Am I supposed to use iteration to find the general formula?

Let's look at the first few terms.
$$a_2=1$$
$$a_4=5+3\cdot 4$$
$$a_8 = 5 a_4 + 3 \cdot 8 = 5(5+3 \cdot 4)+3 \cdot 8 = 5^2+3(5 \cdot 4 + 8)$$
$$a_{16}=5a_8+3 \cdot 16 = 5(5^2+3(5\cdot 4 + 8))+3 \cdot 16=5^3+3(5^2 \cdot 4 + 5 \cdot 8 + 16)$$
From here, you can see that the pattern is
\begin{align*}
a_{2^k}
&= 5^{k-1} + 3 \sum_{j=0}^{k-2} 5^j 2^{k-j}\\
&= 5^{k-1} + 3 \cdot 2^k \cdot \frac{(5/2)^{k-1}-1}{(5/2)-1}\\
&= 5^{k-1} + 2^{k+1}((5/2)^{k-1}-1)\\
&= 5^{k-1}+5^{k-1} \cdot 4 - 2^{k+1}\\
&= 5^k-2^{k+1}.
\end{align*}
