Find x so that $\left(\frac{3}{2}+ \sum_{i=1}^{x} 3^{i}\right)^2$ Find $x$ so that  $$\left(\frac{3}{2}+ \sum_{i=1}^{x} 3^{i}\right)^2 = \frac{3^{22}}{4}$$
I've tried with simpler values for $x$ such as $0, 1$ and $2$. But I can't seem to find any pattern I can take advantage of. How do I solve it? Where would I learn how to solve things like these?

We have that
$$\left(\frac{3}{2}+ \sum_{i=1}^{x} 3^{i}\right)^2 = \frac{3^{22}}{4} \iff \frac{3}{2}+ \sum_{i=1}^{x} 3^{i} = \frac{3^{11}}{2} \iff \sum_{i=1}^{x} 3^{i} = \frac{3^{11}-3}{2}$$
then use that
$$\sum_{i=1}^{x} a^{i}=\frac{a^{x+1}-1}{a-1}-1$$
