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?

$\dfrac{3^{22}}{4} = \left(\dfrac{3^{11}}{2}\right)^2$
So we have,
$$\frac{3}{2} + \sum_{i=1}^x 3^{i} = \frac{3^{11}}{2} \Rightarrow \sum_{i=1}^x 3^{i} = \frac{3^{11}-3}{2} \Rightarrow 3\left(\frac{3^\color{red}x-1}{2}\right) =  3\left(\frac{3^\color{red}{10}-1}{2}\right)$$
