The standard equation for the sum of a geometric series is
$\frac{1-r^{n+1}}{1-r}$
where $sum = a + ar + ar^2 + ... + ar^n$
Given another similar function, $s(n)=\frac{1-r^{\frac{n}{2}+1}}{1-r}$ is it possible to find/recover an unknown sequence such that
$s(n)=a_{0}+a_{1}+a_{2}+...+a_{n}$
meaning instead of doing $s(2)$ I am a able to manually add up the components like $a_{0}+a_{1}+a_{2}$ to get $s(2)$?
What I have already tried to do is use a sum law which states that $a_{n}=s_{n}-s_{n-1}$ and then rearrange to figure out what $a_{n}$ is. However this does not seem to work for me in practice because $s(0)$ should equal $a_{0}$, meaning $s(0)-s(-1)=a_{0}=s(0)$ but this does not happen with that equation and they are unequal.
What I'm thinking is that my function $s$ might not actually represent the sum of a geometric series anymore, could this be the case? How might I go about figuring this out?
The $s_n$ is supposed to be $s_n = \sum_{i=0}^n a_n$, and that expression is only defined for $n \ge 0$. The concept of $s_{-1}$ doesn't make sense in this context.
So there is no such value as $s_{-1}$. The formula $$s_n = \dfrac{1-r^{\frac{n}{2}+1}}{1-r}$$ only holds for $n \ge 0$. The formula $a_n = s_n - s_{n-1}$ only holds for $n\ge 1$, and for $a_0$ instead we just have $$a_0 = s_0$$