What's the easiest way to find if this series is monotonic? This is my sequence:
$$a_n=\frac{3^n-4^n}{2^n-n+1}$$
So I am supposed to check how it is monotonic. Thats what i tried to run using wolfram, but honestly it gets complicated easily
$$\frac{3^{n+1}-4^{n+1}}{2^{n+1}-n}-\frac{3^n-4^n}{2^n-n+1}$$
I honestly don't know what to do after altering forms

Let consider 
$$a_n=\frac{3^n-4^n}{2^n-n+1}=\frac{3^n-4^n}{4^n}\cdot \frac{4^n}{2^n-n+1}=b_n \cdot c_n$$
and
$$b_n=\frac{3^n-4^n}{4^n}=\left(\frac34\right)^n-1$$
is negative strictly decreasing while
$$c_n=\frac{4^n}{2^n-n+1}$$
is positive strictly increasing indeed
$$\frac{4^{n+1}}{2^{n+1}-n}-\frac{4^n}{2^n-n+1}=4^n\left(\frac{4}{2^{n+1}-n}-\frac{1}{2^n-n+1}\right)>0$$
and 
$$\frac{4}{2^{n+1}-n}-\frac{1}{2^n-n+1}>0\iff 4(2^n-n+1)> 2^{n+1}-n$$
$$2^{n+2}-2^{n+1}-3n+4>0 \iff 2^{n+1}-3n+4>0$$
which is true, therefore the product $b_n \cdot c_n$ is strictly decreasing.
