trying to find the general term for a serie in an easy way I have a general expression which is the following:
$$b_0=7$$
$$b_n = 2b_{n−1}+7\cdot4^n\quad∀n∈\Bbb N^+$$
and that I have to it resolve in the easiest way possible. I know that I could use the generative function technique, but there is probably quicker ways.
How could I resolve this? 

Here's another way to look at this. Consider the general form
$$f_n=A\cdot B^n+Cf_{n-1}$$
Then we can write
$$\frac{f_n-Cf_{n-1}}{f_{n-1}-Cf_{n-2}}=\frac{A\cdot B^n}{A\cdot B^{n-1}}=B\\
$$
or
$$
f_n=af_{n-1}+bf_{n-2}\\
$$
where
$$
a=B+C\\
b=-B
$$
This is now in a familiar form for which we know the characteristic roots
$$\alpha,\beta=\frac{a\pm \sqrt{a^2+4b}}{2}$$
And we'll also need $f_1=A\cdot B+ Cf_0$ to complete the solution.
