<s>[INST] Calculating Imaginary Number Is this step-by-step computation correct?
\begin{align}
\frac {3i^{30}-i^{19}}{2i-1} &= \frac {3(i^{2})^{15}-(i^{2})^{9}i}{2i-1} \\
&= \frac {3(-1)^{15}-(-1)^{9}i}{2i-1} \\
&= \frac {-3+i}{2i-1} \\
&= \frac {-3+i}{2i-1} \times \frac {2i+1}{2i+1} \\
&= \frac {-6i-5+i}{-5} \\
&= \frac {-5i-5}{-5} \\
&= i+1.
\end{align}
 [/INST] An alternate method is to take advantage of $i^{4} = 1$ in such a way that $i^{19} = i^{4 \cdot 4 + 3} = (i^{4})^{4} \, i^{3} = i^{3} = - i$ and $i^{30} = i^{4 \cdot 7 + 2} = i^{2} =-1$. This leads to
\begin{align}
\frac{3i^{30}-i^{19}}{2i-1} &= \frac{i - 3}{2i -1} = \frac{(i-3)(2i +1)}{4 i^{2} - 1} = i+1. 
\end{align}
</s>