What equation would match this series? [closed]

The sequence you describe is $a_n = 1 + \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2^{n-1}}$, or written more succintcly, $a_n = \sum_{k=0}^{n-1} \frac{1}{2^k}$.

Note that it is a sequence and not a series (it is, however, the sequence of partial sums of the series $\sum_{k=0}^\infty \frac{1}{2^k}$).

You ask for the 'set of solutions' of this sequence, which does not make sense because an equation has a solution, a sequence does not. I can only guess that you mean to find a simpler formula for $a_n$. Since $a_n$ is a partial sum of a geometric series with quotient $\frac{1}{2}$ we have by the well known formula $a_n = \frac{1-(\frac{1}{2})^n}{1-\frac{1}{2}} = 2 - (\frac{1}{2})^{n-1}$ for $n\in\mathbb{N}$, and this approaches $2$ as $n\to\infty$.

The general formula is that $\sum_{k=0}^\infty q^n = \frac{1}{1-q}$ for every $-1<q<1$. So for $q=\frac{1}{2}$ we have that the limit is $2$, for $q=0.6$ the limit is $\frac{1}{1-0.6}=2.5$, and so on.

A proof of this formula can be found here: https://en.wikipedia.org/wiki/Geometric_series#Sum