So for questions like this, we are given a sum of a sequence and have to show that its geometric progression. We start by finding a formula for $u_n$.
So as per the picture, we start by showing that $u_n$=$s_n-s_{n-1}$ to get a formula that is true for $n \ge 2$. It does not hold for $u_1$ because $s_1-s_0$ does not make sense in the context of the question and thus we have to do $u_1=s_1$ and then check whether the formula for $u_n$ holds for $n \ge 1$. All of this makes sense to me but im trying to think of a counterexample where if we do not check for the $u_1$ case, its possible for the series to not be geometric.